{"id":2694,"date":"2026-01-08T13:53:31","date_gmt":"2026-01-08T08:23:31","guid":{"rendered":"https:\/\/www.manabadi.co.in\/boards\/?p=2694"},"modified":"2026-01-08T16:35:53","modified_gmt":"2026-01-08T11:05:53","slug":"ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3d","status":"publish","type":"post","link":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3d\/","title":{"rendered":"TS Inter 1st Year Maths 1A Matrices Solutions Exercise 3(d)"},"content":{"rendered":"\n<h2 class=\"wp-block-heading\">I.<br> Question 1.<br> Find the determinants of the following matrices.<\/h2>\n\n\n\n<p>(i) <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-1-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-1\" style=\"width: 4.942em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.243em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.773em, 1004.01em, 4.429em, -999.998em); top: -3.353em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-2\"><span class=\"mrow\" id=\"MathJax-Span-3\"><span class=\"mo\" id=\"MathJax-Span-4\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-5\"><span style=\"display: inline-block; position: relative; width: 2.798em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.565em, 1000.47em, 4.895em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -4.658em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-6\"><span class=\"mrow\" id=\"MathJax-Span-7\"><span class=\"mn\" id=\"MathJax-Span-8\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.26em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-12\"><span class=\"mrow\" id=\"MathJax-Span-13\"><span class=\"mn\" id=\"MathJax-Span-14\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1001.21em, 4.988em, -999.998em); top: -4.005em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -4.658em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-9\"><span class=\"mrow\" id=\"MathJax-Span-10\"><span class=\"mn\" id=\"MathJax-Span-11\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -3.26em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-15\"><span class=\"mrow\" id=\"MathJax-Span-16\"><span class=\"mo\" id=\"MathJax-Span-17\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-18\" style=\"font-family: MathJax_Main;\">5<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-19\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">]<\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 3.357em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.132em; border-left: 0px solid; width: 0px; height: 2.922em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-1\">\\left[\\begin{array}{cc}  2 & 1 \\\\  1 & -5  \\end{array}\\right]<\/script><br>\n\n<\/p>\n\n\n\n<p>Answer:<br> Let A = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-2-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-20\" style=\"width: 4.942em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.243em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.773em, 1004.01em, 4.429em, -999.998em); top: -3.353em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-21\"><span class=\"mrow\" id=\"MathJax-Span-22\"><span class=\"mo\" id=\"MathJax-Span-23\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-24\"><span style=\"display: inline-block; position: relative; width: 2.798em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.565em, 1000.47em, 4.895em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -4.658em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-25\"><span class=\"mrow\" id=\"MathJax-Span-26\"><span class=\"mn\" id=\"MathJax-Span-27\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.26em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-31\"><span class=\"mrow\" id=\"MathJax-Span-32\"><span class=\"mn\" id=\"MathJax-Span-33\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1001.21em, 4.988em, -999.998em); top: -4.005em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -4.658em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-28\"><span class=\"mrow\" id=\"MathJax-Span-29\"><span class=\"mn\" id=\"MathJax-Span-30\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -3.26em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-34\"><span class=\"mrow\" id=\"MathJax-Span-35\"><span class=\"mo\" id=\"MathJax-Span-36\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-37\" style=\"font-family: MathJax_Main;\">5<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-38\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">]<\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 3.357em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.132em; border-left: 0px solid; width: 0px; height: 2.922em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-2\">\\left[\\begin{array}{cc}  2 & 1 \\\\  1 & -5  \\end{array}\\right]<\/script> then determinant A<br> = det A = |A| = 2(-5) \u2013 1(1)<br> = -10 \u2013 1<br> = -11<\/p>\n\n\n\n<p>(ii) <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-3-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-39\" style=\"width: 4.942em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.243em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.773em, 1004.01em, 4.429em, -999.998em); top: -3.353em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-40\"><span class=\"mrow\" id=\"MathJax-Span-41\"><span class=\"mo\" id=\"MathJax-Span-42\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-43\"><span style=\"display: inline-block; position: relative; width: 2.798em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.565em, 1001.21em, 4.988em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -4.658em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-44\"><span class=\"mrow\" id=\"MathJax-Span-45\"><span class=\"mn\" id=\"MathJax-Span-46\" style=\"font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -3.26em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-50\"><span class=\"mrow\" id=\"MathJax-Span-51\"><span class=\"mo\" id=\"MathJax-Span-52\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-53\" style=\"font-family: MathJax_Main;\">6<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1000.47em, 4.895em, -999.998em); top: -4.005em; left: 2.286em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -4.658em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-47\"><span class=\"mrow\" id=\"MathJax-Span-48\"><span class=\"mn\" id=\"MathJax-Span-49\" style=\"font-family: MathJax_Main;\">5<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.26em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-54\"><span class=\"mrow\" id=\"MathJax-Span-55\"><span class=\"mn\" id=\"MathJax-Span-56\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-57\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">]<\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 3.357em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.132em; border-left: 0px solid; width: 0px; height: 2.922em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-3\">\\left[\\begin{array}{cc}  4 & 5 \\\\  -6 & 2  \\end{array}\\right]<\/script><br>\n\n<\/p>\n\n\n\n<p>Answer:<br> Let A = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-4-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-58\" style=\"width: 4.942em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.243em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.773em, 1004.01em, 4.429em, -999.998em); top: -3.353em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-59\"><span class=\"mrow\" id=\"MathJax-Span-60\"><span class=\"mo\" id=\"MathJax-Span-61\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-62\"><span style=\"display: inline-block; position: relative; width: 2.798em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.565em, 1001.21em, 4.988em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -4.658em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-63\"><span class=\"mrow\" id=\"MathJax-Span-64\"><span class=\"mn\" id=\"MathJax-Span-65\" style=\"font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -3.26em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-69\"><span class=\"mrow\" id=\"MathJax-Span-70\"><span class=\"mo\" id=\"MathJax-Span-71\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-72\" style=\"font-family: MathJax_Main;\">6<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1000.47em, 4.895em, -999.998em); top: -4.005em; left: 2.286em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -4.658em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-66\"><span class=\"mrow\" id=\"MathJax-Span-67\"><span class=\"mn\" id=\"MathJax-Span-68\" style=\"font-family: MathJax_Main;\">5<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.26em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-73\"><span class=\"mrow\" id=\"MathJax-Span-74\"><span class=\"mn\" id=\"MathJax-Span-75\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-76\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">]<\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 3.357em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.132em; border-left: 0px solid; width: 0px; height: 2.922em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-4\">\\left[\\begin{array}{cc}  4 & 5 \\\\  -6 & 2  \\end{array}\\right]<\/script> then<br> det A = 4(2) \u2013 5(-6)<br> = 8 + 30 = 38<\/p>\n\n\n\n<p>(iii) <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-5-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-77\" style=\"width: 4.662em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.01em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.773em, 1003.78em, 4.429em, -999.998em); top: -3.353em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-78\"><span class=\"mrow\" id=\"MathJax-Span-79\"><span class=\"mo\" id=\"MathJax-Span-80\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-81\"><span style=\"display: inline-block; position: relative; width: 2.565em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.565em, 1000.47em, 4.942em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.24em, 4.15em, -999.998em); top: -4.658em; left: 50%; margin-left: -0.137em;\"><span class=\"mtd\" id=\"MathJax-Span-82\"><span class=\"mrow\" id=\"MathJax-Span-83\"><span class=\"texatom\" id=\"MathJax-Span-84\"><span class=\"mrow\" id=\"MathJax-Span-85\"><span class=\"mi\" id=\"MathJax-Span-86\" style=\"font-family: MathJax_Main;\">i<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.26em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-90\"><span class=\"mrow\" id=\"MathJax-Span-91\"><span class=\"mn\" id=\"MathJax-Span-92\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1001.03em, 4.988em, -999.998em); top: -4.005em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 1.074em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -4.658em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-87\"><span class=\"mrow\" id=\"MathJax-Span-88\"><span class=\"mn\" id=\"MathJax-Span-89\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.03em, 4.243em, -999.998em); top: -3.26em; left: 50%; margin-left: -0.51em;\"><span class=\"mtd\" id=\"MathJax-Span-93\"><span class=\"mrow\" id=\"MathJax-Span-94\"><span class=\"mo\" id=\"MathJax-Span-95\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"texatom\" id=\"MathJax-Span-96\"><span class=\"mrow\" id=\"MathJax-Span-97\"><span class=\"mi\" id=\"MathJax-Span-98\" style=\"font-family: MathJax_Main;\">i<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-99\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">]<\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 3.357em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.132em; border-left: 0px solid; width: 0px; height: 2.922em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-5\">\\left[\\begin{array}{cc}  \\mathrm{i} & 0 \\\\  0 & -\\mathrm{i}  \\end{array}\\right]<\/script><br> Answer:<br> Let A = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-6-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-100\" style=\"width: 4.662em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.01em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.773em, 1003.78em, 4.429em, -999.998em); top: -3.353em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-101\"><span class=\"mrow\" id=\"MathJax-Span-102\"><span class=\"mo\" id=\"MathJax-Span-103\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-104\"><span style=\"display: inline-block; position: relative; width: 2.565em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.565em, 1000.47em, 4.942em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.24em, 4.15em, -999.998em); top: -4.658em; left: 50%; margin-left: -0.137em;\"><span class=\"mtd\" id=\"MathJax-Span-105\"><span class=\"mrow\" id=\"MathJax-Span-106\"><span class=\"texatom\" id=\"MathJax-Span-107\"><span class=\"mrow\" id=\"MathJax-Span-108\"><span class=\"mi\" id=\"MathJax-Span-109\" style=\"font-family: MathJax_Main;\">i<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.26em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-113\"><span class=\"mrow\" id=\"MathJax-Span-114\"><span class=\"mn\" id=\"MathJax-Span-115\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1001.03em, 4.988em, -999.998em); top: -4.005em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 1.074em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -4.658em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-110\"><span class=\"mrow\" id=\"MathJax-Span-111\"><span class=\"mn\" id=\"MathJax-Span-112\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.03em, 4.243em, -999.998em); top: -3.26em; left: 50%; margin-left: -0.51em;\"><span class=\"mtd\" id=\"MathJax-Span-116\"><span class=\"mrow\" id=\"MathJax-Span-117\"><span class=\"mo\" id=\"MathJax-Span-118\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"texatom\" id=\"MathJax-Span-119\"><span class=\"mrow\" id=\"MathJax-Span-120\"><span class=\"mi\" id=\"MathJax-Span-121\" style=\"font-family: MathJax_Main;\">i<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-122\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">]<\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 3.357em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.132em; border-left: 0px solid; width: 0px; height: 2.922em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-6\">\\left[\\begin{array}{cc}  \\mathrm{i} & 0 \\\\  0 & -\\mathrm{i}  \\end{array}\\right]<\/script> then<br> det A = i(-1) \u2013 0 = -i<sup>2<\/sup> = 1 (\u2235 i<sup>2<\/sup> = -1)<\/p>\n\n\n\n<p>(iv) <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-7-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-123\" style=\"width: 6.014em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 5.175em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1004.85em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-124\"><span class=\"mrow\" id=\"MathJax-Span-125\"><span class=\"mo\" id=\"MathJax-Span-126\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Size4; top: -2.84em; left: 0em;\">\u23a1<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Size4; top: -0.836em; left: 0em;\">\u23a3<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Size4; position: absolute; top: -1.815em; left: 0em;\">\u23a2<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-127\"><span style=\"display: inline-block; position: relative; width: 3.497em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.06em, -999.998em); top: -4.471em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-128\"><span class=\"mrow\" id=\"MathJax-Span-129\"><span class=\"mn\" id=\"MathJax-Span-130\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-137\"><span class=\"mrow\" id=\"MathJax-Span-138\"><span class=\"mn\" id=\"MathJax-Span-139\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-146\"><span class=\"mrow\" id=\"MathJax-Span-147\"><span class=\"mn\" id=\"MathJax-Span-148\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.06em, -999.998em); top: -4.471em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-131\"><span class=\"mrow\" id=\"MathJax-Span-132\"><span class=\"mn\" id=\"MathJax-Span-133\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-140\"><span class=\"mrow\" id=\"MathJax-Span-141\"><span class=\"mn\" id=\"MathJax-Span-142\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-149\"><span class=\"mrow\" id=\"MathJax-Span-150\"><span class=\"mn\" id=\"MathJax-Span-151\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.107em, -999.998em); top: -4.471em; left: 2.985em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-134\"><span class=\"mrow\" id=\"MathJax-Span-135\"><span class=\"mn\" id=\"MathJax-Span-136\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-143\"><span class=\"mrow\" id=\"MathJax-Span-144\"><span class=\"mn\" id=\"MathJax-Span-145\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-152\"><span class=\"mrow\" id=\"MathJax-Span-153\"><span class=\"mn\" id=\"MathJax-Span-154\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-155\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Size4; top: -2.84em; left: 0em;\">\u23a4<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Size4; top: -0.836em; left: 0em;\">\u23a6<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Size4; position: absolute; top: -1.815em; left: 0em;\">\u23a5<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-7\">\\left[\\begin{array}{lll}  0 & 1 & 1 \\\\  1 & 0 & 1 \\\\  1 & 1 & 0  \\end{array}\\right]<\/script><br>\n\nAnswer:<br> <img fetchpriority=\"high\" decoding=\"async\" class=\"alignnone size-full wp-image-5970\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3d-1.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(d) 1\" width=\"299\" height=\"202\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(d) 1\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(d)\"><\/p>\n\n\n\n<p>(v) <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-8-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-156\" style=\"width: 6.992em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 6.014em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1005.87em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-157\"><span class=\"mrow\" id=\"MathJax-Span-158\"><span class=\"mo\" id=\"MathJax-Span-159\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-160\"><span style=\"display: inline-block; position: relative; width: 5.082em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.425em, 1001.26em, 6.247em, -999.998em); top: -4.564em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-161\"><span class=\"mrow\" id=\"MathJax-Span-162\"><span class=\"mn\" id=\"MathJax-Span-163\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-170\"><span class=\"mrow\" id=\"MathJax-Span-171\"><span class=\"mn\" id=\"MathJax-Span-172\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.26em, 4.243em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-180\"><span class=\"mrow\" id=\"MathJax-Span-181\"><span class=\"mo\" id=\"MathJax-Span-182\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-183\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.379em, 1001.21em, 6.153em, -999.998em); top: -4.518em; left: 2.286em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-164\"><span class=\"mrow\" id=\"MathJax-Span-165\"><span class=\"mn\" id=\"MathJax-Span-166\" style=\"font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-173\"><span class=\"mrow\" id=\"MathJax-Span-174\"><span class=\"mo\" id=\"MathJax-Span-175\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-176\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-184\"><span class=\"mrow\" id=\"MathJax-Span-185\"><span class=\"mn\" id=\"MathJax-Span-186\" style=\"font-family: MathJax_Main;\">7<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.522em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.107em, -999.998em); top: -4.471em; left: 4.569em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-167\"><span class=\"mrow\" id=\"MathJax-Span-168\"><span class=\"mn\" id=\"MathJax-Span-169\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-177\"><span class=\"mrow\" id=\"MathJax-Span-178\"><span class=\"mn\" id=\"MathJax-Span-179\" style=\"font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-187\"><span class=\"mrow\" id=\"MathJax-Span-188\"><span class=\"mn\" id=\"MathJax-Span-189\" style=\"font-family: MathJax_Main;\">6<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-190\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-8\">\\left|\\begin{array}{rrr}  1 & 4 & 2 \\\\  2 & -1 & 4 \\\\  -3 & 7 & 6  \\end{array}\\right|<\/script><br> Answer:<br> Let A = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-9-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-191\" style=\"width: 7.831em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 6.759em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1006.43em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-192\"><span class=\"mrow\" id=\"MathJax-Span-193\"><span class=\"mo\" id=\"MathJax-Span-194\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Size4; top: -2.84em; left: 0em;\">\u23a1<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Size4; top: -0.836em; left: 0em;\">\u23a3<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Size4; position: absolute; top: -1.815em; left: 0em;\">\u23a2<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-195\"><span style=\"display: inline-block; position: relative; width: 5.082em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.425em, 1001.26em, 6.247em, -999.998em); top: -4.564em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-196\"><span class=\"mrow\" id=\"MathJax-Span-197\"><span class=\"mn\" id=\"MathJax-Span-198\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-205\"><span class=\"mrow\" id=\"MathJax-Span-206\"><span class=\"mn\" id=\"MathJax-Span-207\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.26em, 4.243em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-215\"><span class=\"mrow\" id=\"MathJax-Span-216\"><span class=\"mo\" id=\"MathJax-Span-217\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-218\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.379em, 1001.21em, 6.153em, -999.998em); top: -4.518em; left: 2.286em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-199\"><span class=\"mrow\" id=\"MathJax-Span-200\"><span class=\"mn\" id=\"MathJax-Span-201\" style=\"font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-208\"><span class=\"mrow\" id=\"MathJax-Span-209\"><span class=\"mo\" id=\"MathJax-Span-210\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-211\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-219\"><span class=\"mrow\" id=\"MathJax-Span-220\"><span class=\"mn\" id=\"MathJax-Span-221\" style=\"font-family: MathJax_Main;\">7<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.522em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.107em, -999.998em); top: -4.471em; left: 4.569em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-202\"><span class=\"mrow\" id=\"MathJax-Span-203\"><span class=\"mn\" id=\"MathJax-Span-204\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-212\"><span class=\"mrow\" id=\"MathJax-Span-213\"><span class=\"mn\" id=\"MathJax-Span-214\" style=\"font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-222\"><span class=\"mrow\" id=\"MathJax-Span-223\"><span class=\"mn\" id=\"MathJax-Span-224\" style=\"font-family: MathJax_Main;\">6<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-225\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Size4; top: -2.84em; left: 0em;\">\u23a4<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Size4; top: -0.836em; left: 0em;\">\u23a6<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Size4; position: absolute; top: -1.815em; left: 0em;\">\u23a5<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-9\">\\left[\\begin{array}{rrr}  1 & 4 & 2 \\\\  2 & -1 & 4 \\\\  -3 & 7 & 6  \\end{array}\\right]<\/script><br> Then det A = 1<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-10-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-226\" style=\"width: 4.336em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.73em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1003.59em, 4.802em, -999.998em); top: -3.726em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-227\"><span class=\"mrow\" id=\"MathJax-Span-228\"><span class=\"mo\" id=\"MathJax-Span-229\" style=\"vertical-align: 1.447em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -1.862em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.561em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-230\"><span style=\"display: inline-block; position: relative; width: 2.798em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.565em, 1001.21em, 4.942em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -4.658em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-231\"><span class=\"mrow\" id=\"MathJax-Span-232\"><span class=\"mo\" id=\"MathJax-Span-233\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-234\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -3.26em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-238\"><span class=\"mrow\" id=\"MathJax-Span-239\"><span class=\"mn\" id=\"MathJax-Span-240\" style=\"font-family: MathJax_Main;\">7<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1000.47em, 4.942em, -999.998em); top: -4.005em; left: 2.286em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -4.658em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-235\"><span class=\"mrow\" id=\"MathJax-Span-236\"><span class=\"mn\" id=\"MathJax-Span-237\" style=\"font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.26em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-241\"><span class=\"mrow\" id=\"MathJax-Span-242\"><span class=\"mn\" id=\"MathJax-Span-243\" style=\"font-family: MathJax_Main;\">6<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-244\" style=\"vertical-align: 1.447em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -1.862em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.561em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 3.73em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.132em; border-left: 0px solid; width: 0px; height: 2.922em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-10\">\\left|\\begin{array}{rr}  -1 & 4 \\\\  7 & 6  \\end{array}\\right|<\/script> \u2013 4<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-11-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-245\" style=\"width: 4.336em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.73em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1003.59em, 4.802em, -999.998em); top: -3.726em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-246\"><span class=\"mrow\" id=\"MathJax-Span-247\"><span class=\"mo\" id=\"MathJax-Span-248\" style=\"vertical-align: 1.447em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -1.862em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.561em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-249\"><span style=\"display: inline-block; position: relative; width: 2.798em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.565em, 1001.26em, 4.988em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-250\"><span class=\"mrow\" id=\"MathJax-Span-251\"><span class=\"mo\" id=\"MathJax-Span-252\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-253\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.26em, 4.243em, -999.998em); top: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-257\"><span class=\"mrow\" id=\"MathJax-Span-258\"><span class=\"mo\" id=\"MathJax-Span-259\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-260\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1000.47em, 4.942em, -999.998em); top: -4.005em; left: 2.286em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-254\"><span class=\"mrow\" id=\"MathJax-Span-255\"><span class=\"mn\" id=\"MathJax-Span-256\" style=\"font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-261\"><span class=\"mrow\" id=\"MathJax-Span-262\"><span class=\"mn\" id=\"MathJax-Span-263\" style=\"font-family: MathJax_Main;\">6<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-264\" style=\"vertical-align: 1.447em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -1.862em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.561em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 3.73em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.132em; border-left: 0px solid; width: 0px; height: 2.922em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-11\">\\left|\\begin{array}{ll}  -2 & 4 \\\\  -3 & 6  \\end{array}\\right|<\/script> + 2<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-12-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-265\" style=\"width: 5.221em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.476em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1004.34em, 4.802em, -999.998em); top: -3.726em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-266\"><span class=\"mrow\" id=\"MathJax-Span-267\"><span class=\"mo\" id=\"MathJax-Span-268\" style=\"vertical-align: 1.447em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -1.862em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.561em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-269\"><span style=\"display: inline-block; position: relative; width: 3.544em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.565em, 1001.26em, 4.988em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -4.658em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-270\"><span class=\"mrow\" id=\"MathJax-Span-271\"><span class=\"mn\" id=\"MathJax-Span-272\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.26em, 4.243em, -999.998em); top: -3.26em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-277\"><span class=\"mrow\" id=\"MathJax-Span-278\"><span class=\"mo\" id=\"MathJax-Span-279\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-280\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1001.21em, 4.942em, -999.998em); top: -4.005em; left: 2.286em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -4.658em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-273\"><span class=\"mrow\" id=\"MathJax-Span-274\"><span class=\"mo\" id=\"MathJax-Span-275\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-276\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -3.26em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-281\"><span class=\"mrow\" id=\"MathJax-Span-282\"><span class=\"mn\" id=\"MathJax-Span-283\" style=\"font-family: MathJax_Main;\">7<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-284\" style=\"vertical-align: 1.447em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -1.862em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.561em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 3.73em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.132em; border-left: 0px solid; width: 0px; height: 2.922em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-12\">\\left|\\begin{array}{rr}  2 & -1 \\\\  -3 & 7  \\end{array}\\right|<\/script><br> = 1(-6 \u2013 28) \u2013 4(12 + 12)+ 2(14 \u2013 3)<br> = 1 (- 34) \u2013 4(24) + 2(11)<br> = -34 \u2013 96 + 22<br> = -108<\/p>\n\n\n\n<p>(vi) <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-13-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-285\" style=\"width: 6.945em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 5.967em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1005.64em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-286\"><span class=\"mrow\" id=\"MathJax-Span-287\"><span class=\"mo\" id=\"MathJax-Span-288\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Size4; top: -2.84em; left: 0em;\">\u23a1<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Size4; top: -0.836em; left: 0em;\">\u23a3<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Size4; position: absolute; top: -1.815em; left: 0em;\">\u23a2<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-289\"><span style=\"display: inline-block; position: relative; width: 4.289em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.06em, -999.998em); top: -4.471em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-290\"><span class=\"mrow\" id=\"MathJax-Span-291\"><span class=\"mn\" id=\"MathJax-Span-292\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-300\"><span class=\"mrow\" id=\"MathJax-Span-301\"><span class=\"mn\" id=\"MathJax-Span-302\" style=\"font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-310\"><span class=\"mrow\" id=\"MathJax-Span-311\"><span class=\"mn\" id=\"MathJax-Span-312\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1001.26em, 6.06em, -999.998em); top: -4.471em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-293\"><span class=\"mrow\" id=\"MathJax-Span-294\"><span class=\"mo\" id=\"MathJax-Span-295\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-296\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.26em, 4.243em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-303\"><span class=\"mrow\" id=\"MathJax-Span-304\"><span class=\"mo\" id=\"MathJax-Span-305\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-306\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-313\"><span class=\"mrow\" id=\"MathJax-Span-314\"><span class=\"mn\" id=\"MathJax-Span-315\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.06em, -999.998em); top: -4.471em; left: 3.777em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-297\"><span class=\"mrow\" id=\"MathJax-Span-298\"><span class=\"mn\" id=\"MathJax-Span-299\" style=\"font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-307\"><span class=\"mrow\" id=\"MathJax-Span-308\"><span class=\"mn\" id=\"MathJax-Span-309\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-316\"><span class=\"mrow\" id=\"MathJax-Span-317\"><span class=\"mn\" id=\"MathJax-Span-318\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-319\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Size4; top: -2.84em; left: 0em;\">\u23a4<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Size4; top: -0.836em; left: 0em;\">\u23a6<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Size4; position: absolute; top: -1.815em; left: 0em;\">\u23a5<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-13\">\\left[\\begin{array}{rrr}  2 & -1 & 4 \\\\  4 & -3 & 1 \\\\  1 & 2 & 1  \\end{array}\\right]<\/script><br>\n\nAnswer:<br> Let A = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-14-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-320\" style=\"width: 6.945em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 5.967em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1005.64em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-321\"><span class=\"mrow\" id=\"MathJax-Span-322\"><span class=\"mo\" id=\"MathJax-Span-323\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Size4; top: -2.84em; left: 0em;\">\u23a1<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Size4; top: -0.836em; left: 0em;\">\u23a3<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Size4; position: absolute; top: -1.815em; left: 0em;\">\u23a2<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-324\"><span style=\"display: inline-block; position: relative; width: 4.289em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.06em, -999.998em); top: -4.471em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-325\"><span class=\"mrow\" id=\"MathJax-Span-326\"><span class=\"mn\" id=\"MathJax-Span-327\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-335\"><span class=\"mrow\" id=\"MathJax-Span-336\"><span class=\"mn\" id=\"MathJax-Span-337\" style=\"font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-345\"><span class=\"mrow\" id=\"MathJax-Span-346\"><span class=\"mn\" id=\"MathJax-Span-347\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1001.26em, 6.06em, -999.998em); top: -4.471em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-328\"><span class=\"mrow\" id=\"MathJax-Span-329\"><span class=\"mo\" id=\"MathJax-Span-330\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-331\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.26em, 4.243em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-338\"><span class=\"mrow\" id=\"MathJax-Span-339\"><span class=\"mo\" id=\"MathJax-Span-340\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-341\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-348\"><span class=\"mrow\" id=\"MathJax-Span-349\"><span class=\"mn\" id=\"MathJax-Span-350\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.06em, -999.998em); top: -4.471em; left: 3.777em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-332\"><span class=\"mrow\" id=\"MathJax-Span-333\"><span class=\"mn\" id=\"MathJax-Span-334\" style=\"font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-342\"><span class=\"mrow\" id=\"MathJax-Span-343\"><span class=\"mn\" id=\"MathJax-Span-344\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-351\"><span class=\"mrow\" id=\"MathJax-Span-352\"><span class=\"mn\" id=\"MathJax-Span-353\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-354\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Size4; top: -2.84em; left: 0em;\">\u23a4<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Size4; top: -0.836em; left: 0em;\">\u23a6<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Size4; position: absolute; top: -1.815em; left: 0em;\">\u23a5<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-14\">\\left[\\begin{array}{rrr}  2 & -1 & 4 \\\\  4 & -3 & 1 \\\\  1 & 2 & 1  \\end{array}\\right]<\/script><br> Then det A = 2<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-15-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-355\" style=\"width: 4.336em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.73em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1003.59em, 4.802em, -999.998em); top: -3.726em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-356\"><span class=\"mrow\" id=\"MathJax-Span-357\"><span class=\"mo\" id=\"MathJax-Span-358\" style=\"vertical-align: 1.447em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -1.862em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.561em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-359\"><span style=\"display: inline-block; position: relative; width: 2.798em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.565em, 1001.26em, 4.895em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.26em, 4.243em, -999.998em); top: -4.658em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-360\"><span class=\"mrow\" id=\"MathJax-Span-361\"><span class=\"mo\" id=\"MathJax-Span-362\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-363\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.26em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-367\"><span class=\"mrow\" id=\"MathJax-Span-368\"><span class=\"mn\" id=\"MathJax-Span-369\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1000.42em, 4.895em, -999.998em); top: -4.005em; left: 2.286em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -4.658em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-364\"><span class=\"mrow\" id=\"MathJax-Span-365\"><span class=\"mn\" id=\"MathJax-Span-366\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.26em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-370\"><span class=\"mrow\" id=\"MathJax-Span-371\"><span class=\"mn\" id=\"MathJax-Span-372\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-373\" style=\"vertical-align: 1.447em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -1.862em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.561em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 3.73em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.132em; border-left: 0px solid; width: 0px; height: 2.922em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-15\">\\left|\\begin{array}{rr}  -3 & 1 \\\\  2 & 1  \\end{array}\\right|<\/script> + 1<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-16-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-374\" style=\"width: 3.404em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 2.938em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1002.8em, 4.802em, -999.998em); top: -3.726em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-375\"><span class=\"mrow\" id=\"MathJax-Span-376\"><span class=\"mo\" id=\"MathJax-Span-377\" style=\"vertical-align: 1.447em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -1.862em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.561em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-378\"><span style=\"display: inline-block; position: relative; width: 2.006em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.565em, 1000.47em, 4.895em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-379\"><span class=\"mrow\" id=\"MathJax-Span-380\"><span class=\"mn\" id=\"MathJax-Span-381\" style=\"font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-385\"><span class=\"mrow\" id=\"MathJax-Span-386\"><span class=\"mn\" id=\"MathJax-Span-387\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1000.42em, 4.895em, -999.998em); top: -4.005em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-382\"><span class=\"mrow\" id=\"MathJax-Span-383\"><span class=\"mn\" id=\"MathJax-Span-384\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-388\"><span class=\"mrow\" id=\"MathJax-Span-389\"><span class=\"mn\" id=\"MathJax-Span-390\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-391\" style=\"vertical-align: 1.447em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -1.862em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.561em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 3.73em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.132em; border-left: 0px solid; width: 0px; height: 2.922em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-16\">\\left|\\begin{array}{ll}  4 & 1 \\\\  1 & 1  \\end{array}\\right|<\/script> + 4<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-17-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-392\" style=\"width: 4.336em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.73em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1003.59em, 4.802em, -999.998em); top: -3.726em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-393\"><span class=\"mrow\" id=\"MathJax-Span-394\"><span class=\"mo\" id=\"MathJax-Span-395\" style=\"vertical-align: 1.447em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -1.862em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.561em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-396\"><span style=\"display: inline-block; position: relative; width: 2.798em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.565em, 1000.47em, 4.895em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -4.658em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-397\"><span class=\"mrow\" id=\"MathJax-Span-398\"><span class=\"mn\" id=\"MathJax-Span-399\" style=\"font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.26em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-404\"><span class=\"mrow\" id=\"MathJax-Span-405\"><span class=\"mn\" id=\"MathJax-Span-406\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1001.26em, 4.895em, -999.998em); top: -4.005em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.26em, 4.243em, -999.998em); top: -4.658em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-400\"><span class=\"mrow\" id=\"MathJax-Span-401\"><span class=\"mo\" id=\"MathJax-Span-402\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-403\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.26em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-407\"><span class=\"mrow\" id=\"MathJax-Span-408\"><span class=\"mn\" id=\"MathJax-Span-409\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-410\" style=\"vertical-align: 1.447em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -1.862em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.561em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 3.73em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.132em; border-left: 0px solid; width: 0px; height: 2.922em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-17\">\\left|\\begin{array}{rr}  4 & -3 \\\\  1 & 2  \\end{array}\\right|<\/script><br> = 2(- 3 \u2013 2)+ 1(4 \u2013 1) + 4(8 + 3)<br> = 2(-5) + 3 + 4(11)<br> = \u2013 10 + 3 + 44<br> = 37<\/p>\n\n\n\n<p>(vii) <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-18-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-411\" style=\"width: 7.831em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 6.759em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1006.43em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-412\"><span class=\"mrow\" id=\"MathJax-Span-413\"><span class=\"mo\" id=\"MathJax-Span-414\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Size4; top: -2.84em; left: 0em;\">\u23a1<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Size4; top: -0.836em; left: 0em;\">\u23a3<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Size4; position: absolute; top: -1.815em; left: 0em;\">\u23a2<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-415\"><span style=\"display: inline-block; position: relative; width: 5.082em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.06em, -999.998em); top: -4.471em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-416\"><span class=\"mrow\" id=\"MathJax-Span-417\"><span class=\"mn\" id=\"MathJax-Span-418\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-426\"><span class=\"mrow\" id=\"MathJax-Span-427\"><span class=\"mn\" id=\"MathJax-Span-428\" style=\"font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-436\"><span class=\"mrow\" id=\"MathJax-Span-437\"><span class=\"mn\" id=\"MathJax-Span-438\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1001.21em, 6.06em, -999.998em); top: -4.471em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-419\"><span class=\"mrow\" id=\"MathJax-Span-420\"><span class=\"mn\" id=\"MathJax-Span-421\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-429\"><span class=\"mrow\" id=\"MathJax-Span-430\"><span class=\"mo\" id=\"MathJax-Span-431\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-432\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-439\"><span class=\"mrow\" id=\"MathJax-Span-440\"><span class=\"mn\" id=\"MathJax-Span-441\" style=\"font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.425em, 1001.26em, 6.247em, -999.998em); top: -4.564em; left: 3.777em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.26em, 4.243em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-422\"><span class=\"mrow\" id=\"MathJax-Span-423\"><span class=\"mo\" id=\"MathJax-Span-424\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-425\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-433\"><span class=\"mrow\" id=\"MathJax-Span-434\"><span class=\"mn\" id=\"MathJax-Span-435\" style=\"font-family: MathJax_Main;\">7<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-442\"><span class=\"mrow\" id=\"MathJax-Span-443\"><span class=\"mo\" id=\"MathJax-Span-444\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-445\" style=\"font-family: MathJax_Main;\">6<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-446\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Size4; top: -2.84em; left: 0em;\">\u23a4<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Size4; top: -0.836em; left: 0em;\">\u23a6<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Size4; position: absolute; top: -1.815em; left: 0em;\">\u23a5<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-18\">\\left[\\begin{array}{rrr}  1 & 2 & -3 \\\\  4 & -1 & 7 \\\\  2 & 4 & -6  \\end{array}\\right]<\/script><br>\n\nAnswer:<br> Let A = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-19-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-447\" style=\"width: 7.831em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 6.759em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1006.43em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-448\"><span class=\"mrow\" id=\"MathJax-Span-449\"><span class=\"mo\" id=\"MathJax-Span-450\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Size4; top: -2.84em; left: 0em;\">\u23a1<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Size4; top: -0.836em; left: 0em;\">\u23a3<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Size4; position: absolute; top: -1.815em; left: 0em;\">\u23a2<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-451\"><span style=\"display: inline-block; position: relative; width: 5.082em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.06em, -999.998em); top: -4.471em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-452\"><span class=\"mrow\" id=\"MathJax-Span-453\"><span class=\"mn\" id=\"MathJax-Span-454\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-462\"><span class=\"mrow\" id=\"MathJax-Span-463\"><span class=\"mn\" id=\"MathJax-Span-464\" style=\"font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-472\"><span class=\"mrow\" id=\"MathJax-Span-473\"><span class=\"mn\" id=\"MathJax-Span-474\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1001.21em, 6.06em, -999.998em); top: -4.471em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-455\"><span class=\"mrow\" id=\"MathJax-Span-456\"><span class=\"mn\" id=\"MathJax-Span-457\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-465\"><span class=\"mrow\" id=\"MathJax-Span-466\"><span class=\"mo\" id=\"MathJax-Span-467\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-468\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-475\"><span class=\"mrow\" id=\"MathJax-Span-476\"><span class=\"mn\" id=\"MathJax-Span-477\" style=\"font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.425em, 1001.26em, 6.247em, -999.998em); top: -4.564em; left: 3.777em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.26em, 4.243em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-458\"><span class=\"mrow\" id=\"MathJax-Span-459\"><span class=\"mo\" id=\"MathJax-Span-460\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-461\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-469\"><span class=\"mrow\" id=\"MathJax-Span-470\"><span class=\"mn\" id=\"MathJax-Span-471\" style=\"font-family: MathJax_Main;\">7<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-478\"><span class=\"mrow\" id=\"MathJax-Span-479\"><span class=\"mo\" id=\"MathJax-Span-480\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-481\" style=\"font-family: MathJax_Main;\">6<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-482\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Size4; top: -2.84em; left: 0em;\">\u23a4<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Size4; top: -0.836em; left: 0em;\">\u23a6<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Size4; position: absolute; top: -1.815em; left: 0em;\">\u23a5<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-19\">\\left[\\begin{array}{rrr}  1 & 2 & -3 \\\\  4 & -1 & 7 \\\\  2 & 4 & -6  \\end{array}\\right]<\/script><br> Then det A = 1<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-20-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-483\" style=\"width: 5.221em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.476em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1004.34em, 4.802em, -999.998em); top: -3.726em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-484\"><span class=\"mrow\" id=\"MathJax-Span-485\"><span class=\"mo\" id=\"MathJax-Span-486\" style=\"vertical-align: 1.447em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -1.862em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.561em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-487\"><span style=\"display: inline-block; position: relative; width: 3.544em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.565em, 1001.21em, 4.895em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -4.658em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-488\"><span class=\"mrow\" id=\"MathJax-Span-489\"><span class=\"mo\" id=\"MathJax-Span-490\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-491\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -3.26em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-495\"><span class=\"mrow\" id=\"MathJax-Span-496\"><span class=\"mn\" id=\"MathJax-Span-497\" style=\"font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1001.21em, 4.988em, -999.998em); top: -4.005em; left: 2.286em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -4.658em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-492\"><span class=\"mrow\" id=\"MathJax-Span-493\"><span class=\"mn\" id=\"MathJax-Span-494\" style=\"font-family: MathJax_Main;\">7<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -3.26em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-498\"><span class=\"mrow\" id=\"MathJax-Span-499\"><span class=\"mo\" id=\"MathJax-Span-500\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-501\" style=\"font-family: MathJax_Main;\">6<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-502\" style=\"vertical-align: 1.447em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -1.862em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.561em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 3.73em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.132em; border-left: 0px solid; width: 0px; height: 2.922em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-20\">\\left|\\begin{array}{rr}  -1 & 7 \\\\  4 & -6  \\end{array}\\right|<\/script> \u2013 2<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-21-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-503\" style=\"width: 4.336em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.73em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1003.59em, 4.802em, -999.998em); top: -3.726em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-504\"><span class=\"mrow\" id=\"MathJax-Span-505\"><span class=\"mo\" id=\"MathJax-Span-506\" style=\"vertical-align: 1.447em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -1.862em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.561em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-507\"><span style=\"display: inline-block; position: relative; width: 2.798em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.565em, 1000.47em, 4.895em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -4.658em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-508\"><span class=\"mrow\" id=\"MathJax-Span-509\"><span class=\"mn\" id=\"MathJax-Span-510\" style=\"font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.26em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-514\"><span class=\"mrow\" id=\"MathJax-Span-515\"><span class=\"mn\" id=\"MathJax-Span-516\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1001.21em, 4.988em, -999.998em); top: -4.005em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -4.658em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-511\"><span class=\"mrow\" id=\"MathJax-Span-512\"><span class=\"mn\" id=\"MathJax-Span-513\" style=\"font-family: MathJax_Main;\">7<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -3.26em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-517\"><span class=\"mrow\" id=\"MathJax-Span-518\"><span class=\"mo\" id=\"MathJax-Span-519\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-520\" style=\"font-family: MathJax_Main;\">6<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-521\" style=\"vertical-align: 1.447em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -1.862em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.561em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 3.73em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.132em; border-left: 0px solid; width: 0px; height: 2.922em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-21\">\\left|\\begin{array}{rr}  4 & 7 \\\\  2 & -6  \\end{array}\\right|<\/script> \u2013 3<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-22-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-522\" style=\"width: 4.336em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.73em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1003.59em, 4.802em, -999.998em); top: -3.726em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-523\"><span class=\"mrow\" id=\"MathJax-Span-524\"><span class=\"mo\" id=\"MathJax-Span-525\" style=\"vertical-align: 1.447em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -1.862em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.561em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-526\"><span style=\"display: inline-block; position: relative; width: 2.798em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.565em, 1000.47em, 4.895em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -4.658em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-527\"><span class=\"mrow\" id=\"MathJax-Span-528\"><span class=\"mn\" id=\"MathJax-Span-529\" style=\"font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.26em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-534\"><span class=\"mrow\" id=\"MathJax-Span-535\"><span class=\"mn\" id=\"MathJax-Span-536\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1001.21em, 4.895em, -999.998em); top: -4.005em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -4.658em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-530\"><span class=\"mrow\" id=\"MathJax-Span-531\"><span class=\"mo\" id=\"MathJax-Span-532\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-533\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -3.26em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-537\"><span class=\"mrow\" id=\"MathJax-Span-538\"><span class=\"mn\" id=\"MathJax-Span-539\" style=\"font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-540\" style=\"vertical-align: 1.447em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -1.862em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.561em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 3.73em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.132em; border-left: 0px solid; width: 0px; height: 2.922em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-22\">\\left|\\begin{array}{rr}  4 & -1 \\\\  2 & 4  \\end{array}\\right|<\/script><br> = 1(6 \u2013 28) \u2013 2(- 24 \u2013 14) \u2013 3(16 + 2)<br> = -22 + 76 \u2013 54 = 0<br> [Note : Since R<sub>1<\/sub> and R<sub>2<\/sub> are proportional, we have det A = 0.]<\/p>\n\n\n\n<p>(viii) <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-23-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-541\" style=\"width: 6.247em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 5.361em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1005.03em, 6.247em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-542\"><span class=\"mrow\" id=\"MathJax-Span-543\"><span class=\"mo\" id=\"MathJax-Span-544\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Size4; top: -2.84em; left: 0em;\">\u23a1<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Size4; top: -0.836em; left: 0em;\">\u23a3<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Size4; position: absolute; top: -1.815em; left: 0em;\">\u23a2<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-545\"><span style=\"display: inline-block; position: relative; width: 3.684em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.519em, 1000.56em, 6.293em, -999.998em); top: -4.471em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.562em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-546\"><span class=\"mrow\" id=\"MathJax-Span-547\"><span class=\"mi\" id=\"MathJax-Span-548\" style=\"font-family: MathJax_Math-italic;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.56em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-555\"><span class=\"mrow\" id=\"MathJax-Span-556\"><span class=\"mi\" id=\"MathJax-Span-557\" style=\"font-family: MathJax_Math-italic;\">h<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.47em, 4.336em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-564\"><span class=\"mrow\" id=\"MathJax-Span-565\"><span class=\"mi\" id=\"MathJax-Span-566\" style=\"font-family: MathJax_Math-italic;\">g<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.519em, 1000.56em, 6.526em, -999.998em); top: -4.704em; left: 1.587em;\"><span style=\"display: inline-block; position: relative; width: 0.562em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.56em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-549\"><span class=\"mrow\" id=\"MathJax-Span-550\"><span class=\"mi\" id=\"MathJax-Span-551\" style=\"font-family: MathJax_Math-italic;\">h<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.42em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-558\"><span class=\"mrow\" id=\"MathJax-Span-559\"><span class=\"mi\" id=\"MathJax-Span-560\" style=\"font-family: MathJax_Math-italic;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.56em, 4.336em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-567\"><span class=\"mrow\" id=\"MathJax-Span-568\"><span class=\"mi\" id=\"MathJax-Span-569\" style=\"font-family: MathJax_Math-italic;\">f<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.049em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.709em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.286em, 1000.56em, 5.827em, -999.998em); top: -4.238em; left: 3.171em;\"><span style=\"display: inline-block; position: relative; width: 0.562em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.47em, 4.336em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-552\"><span class=\"mrow\" id=\"MathJax-Span-553\"><span class=\"mi\" id=\"MathJax-Span-554\" style=\"font-family: MathJax_Math-italic;\">g<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.56em, 4.336em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-561\"><span class=\"mrow\" id=\"MathJax-Span-562\"><span class=\"mi\" id=\"MathJax-Span-563\" style=\"font-family: MathJax_Math-italic;\">f<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.049em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.42em, 4.15em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-570\"><span class=\"mrow\" id=\"MathJax-Span-571\"><span class=\"mi\" id=\"MathJax-Span-572\" style=\"font-family: MathJax_Math-italic;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.243em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-573\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Size4; top: -2.84em; left: 0em;\">\u23a4<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Size4; top: -0.836em; left: 0em;\">\u23a6<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Size4; position: absolute; top: -1.815em; left: 0em;\">\u23a5<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.997em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-23\">\\left[\\begin{array}{lll}  a & h & g \\\\  h & b & f \\\\  g & f & c  \\end{array}\\right]<\/script><br>\n\nAnswer:<br> Let A = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-24-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-574\" style=\"width: 6.247em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 5.361em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1005.03em, 6.247em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-575\"><span class=\"mrow\" id=\"MathJax-Span-576\"><span class=\"mo\" id=\"MathJax-Span-577\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Size4; top: -2.84em; left: 0em;\">\u23a1<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Size4; top: -0.836em; left: 0em;\">\u23a3<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Size4; position: absolute; top: -1.815em; left: 0em;\">\u23a2<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-578\"><span style=\"display: inline-block; position: relative; width: 3.684em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.519em, 1000.56em, 6.293em, -999.998em); top: -4.471em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.562em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-579\"><span class=\"mrow\" id=\"MathJax-Span-580\"><span class=\"mi\" id=\"MathJax-Span-581\" style=\"font-family: MathJax_Math-italic;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.56em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-588\"><span class=\"mrow\" id=\"MathJax-Span-589\"><span class=\"mi\" id=\"MathJax-Span-590\" style=\"font-family: MathJax_Math-italic;\">h<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.47em, 4.336em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-597\"><span class=\"mrow\" id=\"MathJax-Span-598\"><span class=\"mi\" id=\"MathJax-Span-599\" style=\"font-family: MathJax_Math-italic;\">g<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.519em, 1000.56em, 6.526em, -999.998em); top: -4.704em; left: 1.587em;\"><span style=\"display: inline-block; position: relative; width: 0.562em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.56em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-582\"><span class=\"mrow\" id=\"MathJax-Span-583\"><span class=\"mi\" id=\"MathJax-Span-584\" style=\"font-family: MathJax_Math-italic;\">h<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.42em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-591\"><span class=\"mrow\" id=\"MathJax-Span-592\"><span class=\"mi\" id=\"MathJax-Span-593\" style=\"font-family: MathJax_Math-italic;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.56em, 4.336em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-600\"><span class=\"mrow\" id=\"MathJax-Span-601\"><span class=\"mi\" id=\"MathJax-Span-602\" style=\"font-family: MathJax_Math-italic;\">f<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.049em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.709em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.286em, 1000.56em, 5.827em, -999.998em); top: -4.238em; left: 3.171em;\"><span style=\"display: inline-block; position: relative; width: 0.562em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.47em, 4.336em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-585\"><span class=\"mrow\" id=\"MathJax-Span-586\"><span class=\"mi\" id=\"MathJax-Span-587\" style=\"font-family: MathJax_Math-italic;\">g<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.56em, 4.336em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-594\"><span class=\"mrow\" id=\"MathJax-Span-595\"><span class=\"mi\" id=\"MathJax-Span-596\" style=\"font-family: MathJax_Math-italic;\">f<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.049em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.42em, 4.15em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-603\"><span class=\"mrow\" id=\"MathJax-Span-604\"><span class=\"mi\" id=\"MathJax-Span-605\" style=\"font-family: MathJax_Math-italic;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.243em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-606\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Size4; top: -2.84em; left: 0em;\">\u23a4<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Size4; top: -0.836em; left: 0em;\">\u23a6<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Size4; position: absolute; top: -1.815em; left: 0em;\">\u23a5<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.997em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-24\">\\left[\\begin{array}{lll}  a & h & g \\\\  h & b & f \\\\  g & f & c  \\end{array}\\right]<\/script><br> Then det A = a<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-25-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-607\" style=\"width: 3.544em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.031em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1002.89em, 4.802em, -999.998em); top: -3.726em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-608\"><span class=\"mrow\" id=\"MathJax-Span-609\"><span class=\"mo\" id=\"MathJax-Span-610\" style=\"vertical-align: 1.447em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -1.862em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.561em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-611\"><span style=\"display: inline-block; position: relative; width: 2.099em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.519em, 1000.56em, 5.082em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.562em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.42em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-612\"><span class=\"mrow\" id=\"MathJax-Span-613\"><span class=\"mi\" id=\"MathJax-Span-614\" style=\"font-family: MathJax_Math-italic;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.56em, 4.336em, -999.998em); top: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-618\"><span class=\"mrow\" id=\"MathJax-Span-619\"><span class=\"mi\" id=\"MathJax-Span-620\" style=\"font-family: MathJax_Math-italic;\">f<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.049em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.519em, 1000.56em, 4.895em, -999.998em); top: -4.005em; left: 1.54em;\"><span style=\"display: inline-block; position: relative; width: 0.562em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.56em, 4.336em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-615\"><span class=\"mrow\" id=\"MathJax-Span-616\"><span class=\"mi\" id=\"MathJax-Span-617\" style=\"font-family: MathJax_Math-italic;\">f<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.049em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.42em, 4.15em, -999.998em); top: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-621\"><span class=\"mrow\" id=\"MathJax-Span-622\"><span class=\"mi\" id=\"MathJax-Span-623\" style=\"font-family: MathJax_Math-italic;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-624\" style=\"vertical-align: 1.447em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -1.862em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.561em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 3.73em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.132em; border-left: 0px solid; width: 0px; height: 2.922em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-25\">\\left|\\begin{array}{ll}  b & f \\\\  f & c  \\end{array}\\right|<\/script> \u2013 h<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-26-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-625\" style=\"width: 3.404em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 2.938em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1002.8em, 4.802em, -999.998em); top: -3.726em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-626\"><span class=\"mrow\" id=\"MathJax-Span-627\"><span class=\"mo\" id=\"MathJax-Span-628\" style=\"vertical-align: 1.447em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -1.862em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.561em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-629\"><span style=\"display: inline-block; position: relative; width: 2.006em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.519em, 1000.56em, 5.082em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.562em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.56em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-630\"><span class=\"mrow\" id=\"MathJax-Span-631\"><span class=\"texatom\" id=\"MathJax-Span-632\"><span class=\"mrow\" id=\"MathJax-Span-633\"><span class=\"mi\" id=\"MathJax-Span-634\" style=\"font-family: MathJax_Main;\">h<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.47em, 4.336em, -999.998em); top: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-640\"><span class=\"mrow\" id=\"MathJax-Span-641\"><span class=\"texatom\" id=\"MathJax-Span-642\"><span class=\"mrow\" id=\"MathJax-Span-643\"><span class=\"mi\" id=\"MathJax-Span-644\" style=\"font-family: MathJax_Main;\">g<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.519em, 1000.42em, 4.895em, -999.998em); top: -4.005em; left: 1.54em;\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.38em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-635\"><span class=\"mrow\" id=\"MathJax-Span-636\"><span class=\"texatom\" id=\"MathJax-Span-637\"><span class=\"mrow\" id=\"MathJax-Span-638\"><span class=\"mi\" id=\"MathJax-Span-639\" style=\"font-family: MathJax_Main;\">f<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.049em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.42em, 4.15em, -999.998em); top: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-645\"><span class=\"mrow\" id=\"MathJax-Span-646\"><span class=\"texatom\" id=\"MathJax-Span-647\"><span class=\"mrow\" id=\"MathJax-Span-648\"><span class=\"mi\" id=\"MathJax-Span-649\" style=\"font-family: MathJax_Main;\">c<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-650\" style=\"vertical-align: 1.447em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -1.862em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.561em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 3.73em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.132em; border-left: 0px solid; width: 0px; height: 2.922em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-26\">\\left|\\begin{array}{ll}  \\mathrm{h} & \\mathrm{f} \\\\  \\mathrm{g} & \\mathrm{c}  \\end{array}\\right|<\/script> \u2013 g<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-27-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-651\" style=\"width: 3.59em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.078em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1002.94em, 4.802em, -999.998em); top: -3.726em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-652\"><span class=\"mrow\" id=\"MathJax-Span-653\"><span class=\"mo\" id=\"MathJax-Span-654\" style=\"vertical-align: 1.447em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -1.862em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.561em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-655\"><span style=\"display: inline-block; position: relative; width: 2.146em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.519em, 1000.56em, 5.082em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.562em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.56em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-656\"><span class=\"mrow\" id=\"MathJax-Span-657\"><span class=\"mi\" id=\"MathJax-Span-658\" style=\"font-family: MathJax_Math-italic;\">h<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.47em, 4.336em, -999.998em); top: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-662\"><span class=\"mrow\" id=\"MathJax-Span-663\"><span class=\"mi\" id=\"MathJax-Span-664\" style=\"font-family: MathJax_Math-italic;\">g<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.519em, 1000.56em, 5.082em, -999.998em); top: -4.005em; left: 1.587em;\"><span style=\"display: inline-block; position: relative; width: 0.562em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.42em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-659\"><span class=\"mrow\" id=\"MathJax-Span-660\"><span class=\"mi\" id=\"MathJax-Span-661\" style=\"font-family: MathJax_Math-italic;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.56em, 4.336em, -999.998em); top: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-665\"><span class=\"mrow\" id=\"MathJax-Span-666\"><span class=\"mi\" id=\"MathJax-Span-667\" style=\"font-family: MathJax_Math-italic;\">f<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.049em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-668\" style=\"vertical-align: 1.447em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -1.862em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.561em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 3.73em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.132em; border-left: 0px solid; width: 0px; height: 2.922em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-27\">\\left|\\begin{array}{ll}  h & b \\\\  g & f  \\end{array}\\right|<\/script><br> = a(bc \u2013 f<sup>2<\/sup>) \u2013 h(ch \u2013 fg) + g(fh \u2013 bg)<br> = abc \u2013 af<sup>2<\/sup> \u2013 ch<sup>2<\/sup> + fgh + fgh \u2013 bg<sup>2<\/sup><br> = abc + 2fgh \u2013 af<sup>2<\/sup> \u2013 bg<sup>2<\/sup> \u2013 ch<sup>2<\/sup><\/p>\n\n\n\n<p>(x) <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-33-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-795\" style=\"width: 7.551em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 6.479em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1006.15em, 6.433em, -999.998em); top: -4.518em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-796\"><span class=\"mrow\" id=\"MathJax-Span-797\"><span class=\"mo\" id=\"MathJax-Span-798\" style=\"vertical-align: 2.239em;\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Size4; top: -2.84em; left: 0em;\">\u23a1<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Size4; top: -0.65em; left: 0em;\">\u23a3<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Size4; position: absolute; top: -1.675em; left: 0em;\">\u23a2<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-799\"><span style=\"display: inline-block; position: relative; width: 4.802em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.472em, 1000.93em, 6.573em, -999.998em); top: -4.844em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(2.985em, 1000.93em, 4.15em, -999.998em); top: -5.403em; left: 50%; margin-left: -0.464em;\"><span class=\"mtd\" id=\"MathJax-Span-800\"><span class=\"mrow\" id=\"MathJax-Span-801\"><span class=\"msubsup\" id=\"MathJax-Span-802\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mn\" id=\"MathJax-Span-803\" style=\"font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-804\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.985em, 1000.93em, 4.15em, -999.998em); top: -3.912em; left: 50%; margin-left: -0.464em;\"><span class=\"mtd\" id=\"MathJax-Span-815\"><span class=\"mrow\" id=\"MathJax-Span-816\"><span class=\"msubsup\" id=\"MathJax-Span-817\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mn\" id=\"MathJax-Span-818\" style=\"font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-819\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.985em, 1000.93em, 4.15em, -999.998em); top: -2.467em; left: 50%; margin-left: -0.464em;\"><span class=\"mtd\" id=\"MathJax-Span-830\"><span class=\"mrow\" id=\"MathJax-Span-831\"><span class=\"msubsup\" id=\"MathJax-Span-832\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mn\" id=\"MathJax-Span-833\" style=\"font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-834\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.849em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.425em, 1000.93em, 6.479em, -999.998em); top: -4.797em; left: 1.913em;\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(2.985em, 1000.93em, 4.15em, -999.998em); top: -5.403em; left: 50%; margin-left: -0.464em;\"><span class=\"mtd\" id=\"MathJax-Span-805\"><span class=\"mrow\" id=\"MathJax-Span-806\"><span class=\"msubsup\" id=\"MathJax-Span-807\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mn\" id=\"MathJax-Span-808\" style=\"font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-809\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.985em, 1000.93em, 4.15em, -999.998em); top: -3.912em; left: 50%; margin-left: -0.464em;\"><span class=\"mtd\" id=\"MathJax-Span-820\"><span class=\"mrow\" id=\"MathJax-Span-821\"><span class=\"msubsup\" id=\"MathJax-Span-822\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mn\" id=\"MathJax-Span-823\" style=\"font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-824\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.985em, 1000.93em, 4.15em, -999.998em); top: -2.467em; left: 50%; margin-left: -0.464em;\"><span class=\"mtd\" id=\"MathJax-Span-835\"><span class=\"mrow\" id=\"MathJax-Span-836\"><span class=\"msubsup\" id=\"MathJax-Span-837\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mn\" id=\"MathJax-Span-838\" style=\"font-family: MathJax_Main;\">4<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.425em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-839\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.802em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.472em, 1000.93em, 6.573em, -999.998em); top: -4.844em; left: 3.87em;\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(2.985em, 1000.93em, 4.15em, -999.998em); top: -5.403em; left: 50%; margin-left: -0.464em;\"><span class=\"mtd\" id=\"MathJax-Span-810\"><span class=\"mrow\" id=\"MathJax-Span-811\"><span class=\"msubsup\" id=\"MathJax-Span-812\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mn\" id=\"MathJax-Span-813\" style=\"font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-814\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.985em, 1000.93em, 4.15em, -999.998em); top: -3.912em; left: 50%; margin-left: -0.464em;\"><span class=\"mtd\" id=\"MathJax-Span-825\"><span class=\"mrow\" id=\"MathJax-Span-826\"><span class=\"msubsup\" id=\"MathJax-Span-827\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mn\" id=\"MathJax-Span-828\" style=\"font-family: MathJax_Main;\">4<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.425em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-829\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.985em, 1000.93em, 4.15em, -999.998em); top: -2.467em; left: 50%; margin-left: -0.464em;\"><span class=\"mtd\" id=\"MathJax-Span-840\"><span class=\"mrow\" id=\"MathJax-Span-841\"><span class=\"msubsup\" id=\"MathJax-Span-842\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mn\" id=\"MathJax-Span-843\" style=\"font-family: MathJax_Main;\">5<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-844\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.849em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-845\" style=\"vertical-align: 2.239em;\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Size4; top: -2.84em; left: 0em;\">\u23a4<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Size4; top: -0.65em; left: 0em;\">\u23a6<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Size4; position: absolute; top: -1.675em; left: 0em;\">\u23a5<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.522em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -2.105em; border-left: 0px solid; width: 0px; height: 4.759em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-33\">\\left[\\begin{array}{ccc}  1^2 & 2^2 & 3^2 \\\\  2^2 & 3^2 & 4^2 \\\\  3^2 & 4^2 & 5^2  \\end{array}\\right]<\/script><br>\n\nAnswer:<br> Let A = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-34-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-846\" style=\"width: 16.265em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 14.028em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.192em, 1013.7em, 6.479em, -999.998em); top: -4.564em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-847\"><span class=\"mrow\" id=\"MathJax-Span-848\"><span class=\"mo\" id=\"MathJax-Span-849\" style=\"vertical-align: 2.239em;\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Size4; top: -2.84em; left: 0em;\">\u23a1<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Size4; top: -0.65em; left: 0em;\">\u23a3<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Size4; position: absolute; top: -1.675em; left: 0em;\">\u23a2<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-850\"><span style=\"display: inline-block; position: relative; width: 4.802em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.472em, 1000.93em, 6.573em, -999.998em); top: -4.844em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(2.985em, 1000.93em, 4.15em, -999.998em); top: -5.403em; left: 50%; margin-left: -0.464em;\"><span class=\"mtd\" id=\"MathJax-Span-851\"><span class=\"mrow\" id=\"MathJax-Span-852\"><span class=\"msubsup\" id=\"MathJax-Span-853\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mn\" id=\"MathJax-Span-854\" style=\"font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-855\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.985em, 1000.93em, 4.15em, -999.998em); top: -3.912em; left: 50%; margin-left: -0.464em;\"><span class=\"mtd\" id=\"MathJax-Span-866\"><span class=\"mrow\" id=\"MathJax-Span-867\"><span class=\"msubsup\" id=\"MathJax-Span-868\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mn\" id=\"MathJax-Span-869\" style=\"font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-870\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.985em, 1000.93em, 4.15em, -999.998em); top: -2.467em; left: 50%; margin-left: -0.464em;\"><span class=\"mtd\" id=\"MathJax-Span-881\"><span class=\"mrow\" id=\"MathJax-Span-882\"><span class=\"msubsup\" id=\"MathJax-Span-883\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mn\" id=\"MathJax-Span-884\" style=\"font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-885\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.849em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.425em, 1000.93em, 6.479em, -999.998em); top: -4.797em; left: 1.913em;\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(2.985em, 1000.93em, 4.15em, -999.998em); top: -5.403em; left: 50%; margin-left: -0.464em;\"><span class=\"mtd\" id=\"MathJax-Span-856\"><span class=\"mrow\" id=\"MathJax-Span-857\"><span class=\"msubsup\" id=\"MathJax-Span-858\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mn\" id=\"MathJax-Span-859\" style=\"font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-860\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.985em, 1000.93em, 4.15em, -999.998em); top: -3.912em; left: 50%; margin-left: -0.464em;\"><span class=\"mtd\" id=\"MathJax-Span-871\"><span class=\"mrow\" id=\"MathJax-Span-872\"><span class=\"msubsup\" id=\"MathJax-Span-873\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mn\" id=\"MathJax-Span-874\" style=\"font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-875\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.985em, 1000.93em, 4.15em, -999.998em); top: -2.467em; left: 50%; margin-left: -0.464em;\"><span class=\"mtd\" id=\"MathJax-Span-886\"><span class=\"mrow\" id=\"MathJax-Span-887\"><span class=\"msubsup\" id=\"MathJax-Span-888\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mn\" id=\"MathJax-Span-889\" style=\"font-family: MathJax_Main;\">4<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.425em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-890\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.802em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.472em, 1000.93em, 6.573em, -999.998em); top: -4.844em; left: 3.87em;\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(2.985em, 1000.93em, 4.15em, -999.998em); top: -5.403em; left: 50%; margin-left: -0.464em;\"><span class=\"mtd\" id=\"MathJax-Span-861\"><span class=\"mrow\" id=\"MathJax-Span-862\"><span class=\"msubsup\" id=\"MathJax-Span-863\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mn\" id=\"MathJax-Span-864\" style=\"font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-865\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.985em, 1000.93em, 4.15em, -999.998em); top: -3.912em; left: 50%; margin-left: -0.464em;\"><span class=\"mtd\" id=\"MathJax-Span-876\"><span class=\"mrow\" id=\"MathJax-Span-877\"><span class=\"msubsup\" id=\"MathJax-Span-878\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mn\" id=\"MathJax-Span-879\" style=\"font-family: MathJax_Main;\">4<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.425em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-880\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.985em, 1000.93em, 4.15em, -999.998em); top: -2.467em; left: 50%; margin-left: -0.464em;\"><span class=\"mtd\" id=\"MathJax-Span-891\"><span class=\"mrow\" id=\"MathJax-Span-892\"><span class=\"msubsup\" id=\"MathJax-Span-893\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mn\" id=\"MathJax-Span-894\" style=\"font-family: MathJax_Main;\">5<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-895\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.849em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-896\" style=\"vertical-align: 2.239em;\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Size4; top: -2.84em; left: 0em;\">\u23a4<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Size4; top: -0.65em; left: 0em;\">\u23a6<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Size4; position: absolute; top: -1.675em; left: 0em;\">\u23a5<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-897\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"mrow\" id=\"MathJax-Span-898\" style=\"padding-left: 0.282em;\"><span class=\"mo\" id=\"MathJax-Span-899\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Size4; top: -2.84em; left: 0em;\">\u23a1<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Size4; top: -0.836em; left: 0em;\">\u23a3<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Size4; position: absolute; top: -1.815em; left: 0em;\">\u23a2<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-900\"><span style=\"display: inline-block; position: relative; width: 4.522em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.107em, -999.998em); top: -4.471em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-901\"><span class=\"mrow\" id=\"MathJax-Span-902\"><span class=\"mn\" id=\"MathJax-Span-903\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-910\"><span class=\"mrow\" id=\"MathJax-Span-911\"><span class=\"mn\" id=\"MathJax-Span-912\" style=\"font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-919\"><span class=\"mrow\" id=\"MathJax-Span-920\"><span class=\"mn\" id=\"MathJax-Span-921\" style=\"font-family: MathJax_Main;\">9<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.379em, 1000.98em, 6.153em, -999.998em); top: -4.518em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-904\"><span class=\"mrow\" id=\"MathJax-Span-905\"><span class=\"mn\" id=\"MathJax-Span-906\" style=\"font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-913\"><span class=\"mrow\" id=\"MathJax-Span-914\"><span class=\"mn\" id=\"MathJax-Span-915\" style=\"font-family: MathJax_Main;\">9<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.98em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.51em;\"><span class=\"mtd\" id=\"MathJax-Span-922\"><span class=\"mrow\" id=\"MathJax-Span-923\"><span class=\"mn\" id=\"MathJax-Span-924\" style=\"font-family: MathJax_Main;\">16<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.522em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.98em, 6.107em, -999.998em); top: -4.471em; left: 3.497em;\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-907\"><span class=\"mrow\" id=\"MathJax-Span-908\"><span class=\"mn\" id=\"MathJax-Span-909\" style=\"font-family: MathJax_Main;\">9<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.98em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.51em;\"><span class=\"mtd\" id=\"MathJax-Span-916\"><span class=\"mrow\" id=\"MathJax-Span-917\"><span class=\"mn\" id=\"MathJax-Span-918\" style=\"font-family: MathJax_Main;\">16<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.93em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.51em;\"><span class=\"mtd\" id=\"MathJax-Span-925\"><span class=\"mrow\" id=\"MathJax-Span-926\"><span class=\"mn\" id=\"MathJax-Span-927\" style=\"font-family: MathJax_Main;\">25<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-928\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Size4; top: -2.84em; left: 0em;\">\u23a4<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Size4; top: -0.836em; left: 0em;\">\u23a6<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Size4; position: absolute; top: -1.815em; left: 0em;\">\u23a5<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -2.105em; border-left: 0px solid; width: 0px; height: 4.759em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-34\">\\left[\\begin{array}{ccc}  1^2 & 2^2 & 3^2 \\\\  2^2 & 3^2 & 4^2 \\\\  3^2 & 4^2 & 5^2  \\end{array}\\right]=\\left[\\begin{array}{ccc}  1 & 4 & 9 \\\\  4 & 9 & 16 \\\\  9 & 16 & 25  \\end{array}\\right]<\/script><br> Then det A = 1(225 \u2013 256) \u2013 4(100 \u2013 144) + 9(64 \u2013 81)<br> = -31 + 176 \u2013 153 = -8<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Question 2.<br> If A = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-35-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-929\" style=\"width: 6.992em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 6.014em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1005.69em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-930\"><span class=\"mrow\" id=\"MathJax-Span-931\"><span class=\"mo\" id=\"MathJax-Span-932\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Size4; top: -2.84em; left: 0em;\">\u23a1<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Size4; top: -0.836em; left: 0em;\">\u23a3<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Size4; position: absolute; top: -1.815em; left: 0em;\">\u23a2<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-933\"><span style=\"display: inline-block; position: relative; width: 4.336em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.107em, -999.998em); top: -4.471em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-934\"><span class=\"mrow\" id=\"MathJax-Span-935\"><span class=\"mn\" id=\"MathJax-Span-936\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-943\"><span class=\"mrow\" id=\"MathJax-Span-944\"><span class=\"mn\" id=\"MathJax-Span-945\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-952\"><span class=\"mrow\" id=\"MathJax-Span-953\"><span class=\"mn\" id=\"MathJax-Span-954\" style=\"font-family: MathJax_Main;\">5<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.425em, 1001.21em, 6.247em, -999.998em); top: -4.564em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-937\"><span class=\"mrow\" id=\"MathJax-Span-938\"><span class=\"mn\" id=\"MathJax-Span-939\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-946\"><span class=\"mrow\" id=\"MathJax-Span-947\"><span class=\"mn\" id=\"MathJax-Span-948\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-955\"><span class=\"mrow\" id=\"MathJax-Span-956\"><span class=\"mo\" id=\"MathJax-Span-957\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-958\" style=\"font-family: MathJax_Main;\">6<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.51em, 6.06em, -999.998em); top: -4.471em; left: 3.777em;\"><span style=\"display: inline-block; position: relative; width: 0.562em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-940\"><span class=\"mrow\" id=\"MathJax-Span-941\"><span class=\"mn\" id=\"MathJax-Span-942\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-949\"><span class=\"mrow\" id=\"MathJax-Span-950\"><span class=\"mn\" id=\"MathJax-Span-951\" style=\"font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-959\"><span class=\"mrow\" id=\"MathJax-Span-960\"><span class=\"mi\" id=\"MathJax-Span-961\" style=\"font-family: MathJax_Math-italic;\">x<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-962\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Size4; top: -2.84em; left: 0em;\">\u23a4<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Size4; top: -0.836em; left: 0em;\">\u23a6<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Size4; position: absolute; top: -1.815em; left: 0em;\">\u23a5<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-35\">\\left[\\begin{array}{rrr}  1 & 0 & 0 \\\\  2 & 3 & 4 \\\\  5 & -6 & x  \\end{array}\\right]<\/script> and det A = 45 then find x.<\/h3>\n\n\n\n<p>Answer:<br> det A = 45<br> \u21d2 <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-36-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-963\" style=\"width: 6.107em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 5.268em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1005.13em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-964\"><span class=\"mrow\" id=\"MathJax-Span-965\"><span class=\"mo\" id=\"MathJax-Span-966\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-967\"><span style=\"display: inline-block; position: relative; width: 4.336em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.107em, -999.998em); top: -4.471em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-968\"><span class=\"mrow\" id=\"MathJax-Span-969\"><span class=\"mn\" id=\"MathJax-Span-970\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-977\"><span class=\"mrow\" id=\"MathJax-Span-978\"><span class=\"mn\" id=\"MathJax-Span-979\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-986\"><span class=\"mrow\" id=\"MathJax-Span-987\"><span class=\"mn\" id=\"MathJax-Span-988\" style=\"font-family: MathJax_Main;\">5<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.425em, 1001.21em, 6.247em, -999.998em); top: -4.564em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-971\"><span class=\"mrow\" id=\"MathJax-Span-972\"><span class=\"mn\" id=\"MathJax-Span-973\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-980\"><span class=\"mrow\" id=\"MathJax-Span-981\"><span class=\"mn\" id=\"MathJax-Span-982\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-989\"><span class=\"mrow\" id=\"MathJax-Span-990\"><span class=\"mo\" id=\"MathJax-Span-991\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-992\" style=\"font-family: MathJax_Main;\">6<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.51em, 6.06em, -999.998em); top: -4.471em; left: 3.777em;\"><span style=\"display: inline-block; position: relative; width: 0.562em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-974\"><span class=\"mrow\" id=\"MathJax-Span-975\"><span class=\"mn\" id=\"MathJax-Span-976\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-983\"><span class=\"mrow\" id=\"MathJax-Span-984\"><span class=\"mn\" id=\"MathJax-Span-985\" style=\"font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-993\"><span class=\"mrow\" id=\"MathJax-Span-994\"><span class=\"mi\" id=\"MathJax-Span-995\" style=\"font-family: MathJax_Math-italic;\">x<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-996\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-36\">\\left|\\begin{array}{rrr}  1 & 0 & 0 \\\\  2 & 3 & 4 \\\\  5 & -6 & x  \\end{array}\\right|<\/script> = 45<br> \u21d2 1(3x + 24) = 45<br> \u21d2 3x = 21<br> \u21d2 x = 7<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">II.<br> Question 1.<br> Show that <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-37-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-997\" style=\"width: 7.924em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 6.806em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1006.67em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-998\"><span class=\"mrow\" id=\"MathJax-Span-999\"><span class=\"mo\" id=\"MathJax-Span-1000\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-1001\"><span style=\"display: inline-block; position: relative; width: 5.874em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1001.03em, 6.107em, -999.998em); top: -4.518em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 1.074em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.98em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1002\"><span class=\"mrow\" id=\"MathJax-Span-1003\"><span class=\"texatom\" id=\"MathJax-Span-1004\"><span class=\"mrow\" id=\"MathJax-Span-1005\"><span class=\"mi\" id=\"MathJax-Span-1006\" style=\"font-family: MathJax_Main;\">b<\/span><span class=\"mi\" id=\"MathJax-Span-1007\" style=\"font-family: MathJax_Main;\">c<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.93em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1020\"><span class=\"mrow\" id=\"MathJax-Span-1021\"><span class=\"texatom\" id=\"MathJax-Span-1022\"><span class=\"mrow\" id=\"MathJax-Span-1023\"><span class=\"mi\" id=\"MathJax-Span-1024\" style=\"font-family: MathJax_Main;\">c<\/span><span class=\"mi\" id=\"MathJax-Span-1025\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1001.03em, 4.15em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1038\"><span class=\"mrow\" id=\"MathJax-Span-1039\"><span class=\"texatom\" id=\"MathJax-Span-1040\"><span class=\"mrow\" id=\"MathJax-Span-1041\"><span class=\"mi\" id=\"MathJax-Span-1042\" style=\"font-family: MathJax_Main;\">a<\/span><span class=\"mi\" id=\"MathJax-Span-1043\" style=\"font-family: MathJax_Main;\">b<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.522em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.379em, 1002.29em, 6.247em, -999.998em); top: -4.564em; left: 2.053em;\"><span style=\"display: inline-block; position: relative; width: 2.286em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1002.24em, 4.243em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1008\"><span class=\"mrow\" id=\"MathJax-Span-1009\"><span class=\"texatom\" id=\"MathJax-Span-1010\"><span class=\"mrow\" id=\"MathJax-Span-1011\"><span class=\"mi\" id=\"MathJax-Span-1012\" style=\"font-family: MathJax_Main;\">b<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-1013\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"texatom\" id=\"MathJax-Span-1014\" style=\"padding-left: 0.235em;\"><span class=\"mrow\" id=\"MathJax-Span-1015\"><span class=\"mi\" id=\"MathJax-Span-1016\" style=\"font-family: MathJax_Main;\">c<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.264em, 1002.19em, 4.243em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1026\"><span class=\"mrow\" id=\"MathJax-Span-1027\"><span class=\"texatom\" id=\"MathJax-Span-1028\"><span class=\"mrow\" id=\"MathJax-Span-1029\"><span class=\"mi\" id=\"MathJax-Span-1030\" style=\"font-family: MathJax_Main;\">c<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-1031\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"texatom\" id=\"MathJax-Span-1032\" style=\"padding-left: 0.235em;\"><span class=\"mrow\" id=\"MathJax-Span-1033\"><span class=\"mi\" id=\"MathJax-Span-1034\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1002.29em, 4.243em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1044\"><span class=\"mrow\" id=\"MathJax-Span-1045\"><span class=\"texatom\" id=\"MathJax-Span-1046\"><span class=\"mrow\" id=\"MathJax-Span-1047\"><span class=\"mi\" id=\"MathJax-Span-1048\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-1049\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"texatom\" id=\"MathJax-Span-1050\" style=\"padding-left: 0.235em;\"><span class=\"mrow\" id=\"MathJax-Span-1051\"><span class=\"mi\" id=\"MathJax-Span-1052\" style=\"font-family: MathJax_Main;\">b<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.42em, 6.06em, -999.998em); top: -4.471em; left: 5.361em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1017\"><span class=\"mrow\" id=\"MathJax-Span-1018\"><span class=\"mn\" id=\"MathJax-Span-1019\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1035\"><span class=\"mrow\" id=\"MathJax-Span-1036\"><span class=\"mn\" id=\"MathJax-Span-1037\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1053\"><span class=\"mrow\" id=\"MathJax-Span-1054\"><span class=\"mn\" id=\"MathJax-Span-1055\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-1056\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-37\">\\left|\\begin{array}{lll}  \\mathrm{b c} & \\mathrm{b}+\\mathrm{c} & 1 \\\\  \\mathrm{c a} & \\mathrm{c}+\\mathrm{a} & 1 \\\\  \\mathrm{a b} & \\mathrm{a}+\\mathrm{b} & 1  \\end{array}\\right|<\/script> = (a \u2013 b)(b \u2013 c)(c \u2013 a).<\/h3>\n\n\n\n<p>Answer:<br> Operating R<sub>2<\/sub> \u2013 R<sub>1<\/sub>, R<sub>3<\/sub> \u2013 R<sub>1<\/sub>, on the given determinant<br> LHS = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-38-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-1057\" style=\"width: 10.487em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 9.042em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1008.9em, 6.293em, -999.998em); top: -4.471em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-1058\"><span class=\"mrow\" id=\"MathJax-Span-1059\"><span class=\"mo\" id=\"MathJax-Span-1060\" style=\"vertical-align: 2.192em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.37em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.514em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.815em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.069em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-1061\"><span style=\"display: inline-block; position: relative; width: 8.11em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.565em, 1003.31em, 6.619em, -999.998em); top: -4.797em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 3.404em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.84em, 4.15em, -999.998em); top: -5.403em; left: 50%; margin-left: -0.417em;\"><span class=\"mtd\" id=\"MathJax-Span-1062\"><span class=\"mrow\" id=\"MathJax-Span-1063\"><span class=\"mi\" id=\"MathJax-Span-1064\" style=\"font-family: MathJax_Math-italic;\">b<\/span><span class=\"mi\" id=\"MathJax-Span-1065\" style=\"font-family: MathJax_Math-italic;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.124em, 1003.31em, 4.383em, -999.998em); top: -4.005em; left: 50%; margin-left: -1.722em;\"><span class=\"mtd\" id=\"MathJax-Span-1074\"><span class=\"mrow\" id=\"MathJax-Span-1075\"><span class=\"mi\" id=\"MathJax-Span-1076\" style=\"font-family: MathJax_Math-italic;\">c<\/span><span class=\"mo\" id=\"MathJax-Span-1077\" style=\"font-family: MathJax_Main;\">(<\/span><span class=\"mi\" id=\"MathJax-Span-1078\" style=\"font-family: MathJax_Math-italic;\">a<\/span><span class=\"mo\" id=\"MathJax-Span-1079\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"mi\" id=\"MathJax-Span-1080\" style=\"font-family: MathJax_Math-italic; padding-left: 0.235em;\">b<\/span><span class=\"mo\" id=\"MathJax-Span-1081\" style=\"font-family: MathJax_Main;\">)<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.124em, 1003.31em, 4.383em, -999.998em); top: -2.561em; left: 50%; margin-left: -1.722em;\"><span class=\"mtd\" id=\"MathJax-Span-1090\"><span class=\"mrow\" id=\"MathJax-Span-1091\"><span class=\"mi\" id=\"MathJax-Span-1092\" style=\"font-family: MathJax_Math-italic;\">b<\/span><span class=\"mo\" id=\"MathJax-Span-1093\" style=\"font-family: MathJax_Main;\">(<\/span><span class=\"mi\" id=\"MathJax-Span-1094\" style=\"font-family: MathJax_Math-italic;\">a<\/span><span class=\"mo\" id=\"MathJax-Span-1095\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"mi\" id=\"MathJax-Span-1096\" style=\"font-family: MathJax_Math-italic; padding-left: 0.235em;\">c<\/span><span class=\"mo\" id=\"MathJax-Span-1097\" style=\"font-family: MathJax_Main;\">)<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.802em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.379em, 1002.19em, 6.293em, -999.998em); top: -4.611em; left: 4.429em;\"><span style=\"display: inline-block; position: relative; width: 2.192em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1002.1em, 4.243em, -999.998em); top: -5.403em; left: 50%; margin-left: -1.069em;\"><span class=\"mtd\" id=\"MathJax-Span-1066\"><span class=\"mrow\" id=\"MathJax-Span-1067\"><span class=\"mi\" id=\"MathJax-Span-1068\" style=\"font-family: MathJax_Math-italic;\">b<\/span><span class=\"mo\" id=\"MathJax-Span-1069\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mi\" id=\"MathJax-Span-1070\" style=\"font-family: MathJax_Math-italic; padding-left: 0.235em;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1002.19em, 4.243em, -999.998em); top: -4.005em; left: 50%; margin-left: -1.116em;\"><span class=\"mtd\" id=\"MathJax-Span-1082\"><span class=\"mrow\" id=\"MathJax-Span-1083\"><span class=\"mi\" id=\"MathJax-Span-1084\" style=\"font-family: MathJax_Math-italic;\">a<\/span><span class=\"mo\" id=\"MathJax-Span-1085\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"mi\" id=\"MathJax-Span-1086\" style=\"font-family: MathJax_Math-italic; padding-left: 0.235em;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.264em, 1002.19em, 4.243em, -999.998em); top: -2.561em; left: 50%; margin-left: -1.116em;\"><span class=\"mtd\" id=\"MathJax-Span-1098\"><span class=\"mrow\" id=\"MathJax-Span-1099\"><span class=\"mi\" id=\"MathJax-Span-1100\" style=\"font-family: MathJax_Math-italic;\">a<\/span><span class=\"mo\" id=\"MathJax-Span-1101\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"mi\" id=\"MathJax-Span-1102\" style=\"font-family: MathJax_Math-italic; padding-left: 0.235em;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.616em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.153em, -999.998em); top: -4.518em; left: 7.644em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.403em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-1071\"><span class=\"mrow\" id=\"MathJax-Span-1072\"><span class=\"mn\" id=\"MathJax-Span-1073\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -4.005em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-1087\"><span class=\"mrow\" id=\"MathJax-Span-1088\"><span class=\"mn\" id=\"MathJax-Span-1089\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-1103\"><span class=\"mrow\" id=\"MathJax-Span-1104\"><span class=\"mn\" id=\"MathJax-Span-1105\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.522em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-1106\" style=\"vertical-align: 2.192em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.37em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.514em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.815em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.069em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.997em; border-left: 0px solid; width: 0px; height: 4.651em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-38\">\\left|\\begin{array}{ccc}  b c & b+c & 1 \\\\  c(a-b) & a-b & 0 \\\\  b(a-c) & a-c & 0  \\end{array}\\right|<\/script><br> = (a \u2013 b)(a \u2013 c)<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-39-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-1107\" style=\"width: 7.411em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 6.386em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1006.25em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-1108\"><span class=\"mrow\" id=\"MathJax-Span-1109\"><span class=\"mo\" id=\"MathJax-Span-1110\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-1111\"><span style=\"display: inline-block; position: relative; width: 5.454em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.84em, 6.107em, -999.998em); top: -4.518em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.841em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.84em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.417em;\"><span class=\"mtd\" id=\"MathJax-Span-1112\"><span class=\"mrow\" id=\"MathJax-Span-1113\"><span class=\"mi\" id=\"MathJax-Span-1114\" style=\"font-family: MathJax_Math-italic;\">b<\/span><span class=\"mi\" id=\"MathJax-Span-1115\" style=\"font-family: MathJax_Math-italic;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.42em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-1124\"><span class=\"mrow\" id=\"MathJax-Span-1125\"><span class=\"mi\" id=\"MathJax-Span-1126\" style=\"font-family: MathJax_Math-italic;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.42em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-1133\"><span class=\"mrow\" id=\"MathJax-Span-1134\"><span class=\"mi\" id=\"MathJax-Span-1135\" style=\"font-family: MathJax_Math-italic;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.522em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.286em, 1002.1em, 6.06em, -999.998em); top: -4.471em; left: 1.866em;\"><span style=\"display: inline-block; position: relative; width: 2.099em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1002.1em, 4.243em, -999.998em); top: -5.356em; left: 50%; margin-left: -1.069em;\"><span class=\"mtd\" id=\"MathJax-Span-1116\"><span class=\"mrow\" id=\"MathJax-Span-1117\"><span class=\"mi\" id=\"MathJax-Span-1118\" style=\"font-family: MathJax_Math-italic;\">b<\/span><span class=\"mo\" id=\"MathJax-Span-1119\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mi\" id=\"MathJax-Span-1120\" style=\"font-family: MathJax_Math-italic; padding-left: 0.235em;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-1127\"><span class=\"mrow\" id=\"MathJax-Span-1128\"><span class=\"mn\" id=\"MathJax-Span-1129\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-1136\"><span class=\"mrow\" id=\"MathJax-Span-1137\"><span class=\"mn\" id=\"MathJax-Span-1138\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.107em, -999.998em); top: -4.471em; left: 4.988em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-1121\"><span class=\"mrow\" id=\"MathJax-Span-1122\"><span class=\"mn\" id=\"MathJax-Span-1123\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-1130\"><span class=\"mrow\" id=\"MathJax-Span-1131\"><span class=\"mn\" id=\"MathJax-Span-1132\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-1139\"><span class=\"mrow\" id=\"MathJax-Span-1140\"><span class=\"mn\" id=\"MathJax-Span-1141\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-1142\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-39\">\\left|\\begin{array}{ccc}  b c & b+c & 1 \\\\  c & 1 & 0 \\\\  b & 1 & 0  \\end{array}\\right|<\/script><br> = (a \u2013 b)(a \u2013 c)(1)(c \u2013 b)<br> = (a \u2013 b)(b \u2013 c)(c \u2013 a) (exponding on 3rd column)<br> = RHS<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Question 2.<br> Show that <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-40-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-1143\" style=\"width: 11.093em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 9.555em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1009.41em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-1144\"><span class=\"mrow\" id=\"MathJax-Span-1145\"><span class=\"mo\" id=\"MathJax-Span-1146\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-1147\"><span style=\"display: inline-block; position: relative; width: 8.623em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1002.19em, 6.107em, -999.998em); top: -4.518em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 2.192em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1002.1em, 4.243em, -999.998em); top: -5.356em; left: 50%; margin-left: -1.069em;\"><span class=\"mtd\" id=\"MathJax-Span-1148\"><span class=\"mrow\" id=\"MathJax-Span-1149\"><span class=\"mi\" id=\"MathJax-Span-1150\" style=\"font-family: MathJax_Math-italic;\">b<\/span><span class=\"mo\" id=\"MathJax-Span-1151\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mi\" id=\"MathJax-Span-1152\" style=\"font-family: MathJax_Math-italic; padding-left: 0.235em;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1002.19em, 4.243em, -999.998em); top: -3.959em; left: 50%; margin-left: -1.116em;\"><span class=\"mtd\" id=\"MathJax-Span-1163\"><span class=\"mrow\" id=\"MathJax-Span-1164\"><span class=\"mi\" id=\"MathJax-Span-1165\" style=\"font-family: MathJax_Math-italic;\">a<\/span><span class=\"mo\" id=\"MathJax-Span-1166\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mi\" id=\"MathJax-Span-1167\" style=\"font-family: MathJax_Math-italic; padding-left: 0.235em;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.277em;\"><span class=\"mtd\" id=\"MathJax-Span-1178\"><span class=\"mrow\" id=\"MathJax-Span-1179\"><span class=\"mi\" id=\"MathJax-Span-1180\" style=\"font-family: MathJax_Math-italic;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.522em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1002.19em, 5.967em, -999.998em); top: -4.378em; left: 3.218em;\"><span style=\"display: inline-block; position: relative; width: 2.192em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.264em, 1002.19em, 4.243em, -999.998em); top: -5.356em; left: 50%; margin-left: -1.116em;\"><span class=\"mtd\" id=\"MathJax-Span-1153\"><span class=\"mrow\" id=\"MathJax-Span-1154\"><span class=\"mi\" id=\"MathJax-Span-1155\" style=\"font-family: MathJax_Math-italic;\">c<\/span><span class=\"mo\" id=\"MathJax-Span-1156\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mi\" id=\"MathJax-Span-1157\" style=\"font-family: MathJax_Math-italic; padding-left: 0.235em;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1002.1em, 4.243em, -999.998em); top: -3.959em; left: 50%; margin-left: -1.069em;\"><span class=\"mtd\" id=\"MathJax-Span-1168\"><span class=\"mrow\" id=\"MathJax-Span-1169\"><span class=\"mi\" id=\"MathJax-Span-1170\" style=\"font-family: MathJax_Math-italic;\">b<\/span><span class=\"mo\" id=\"MathJax-Span-1171\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mi\" id=\"MathJax-Span-1172\" style=\"font-family: MathJax_Math-italic; padding-left: 0.235em;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.42em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-1181\"><span class=\"mrow\" id=\"MathJax-Span-1182\"><span class=\"mi\" id=\"MathJax-Span-1183\" style=\"font-family: MathJax_Math-italic;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.383em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1002.19em, 6.107em, -999.998em); top: -4.518em; left: 6.433em;\"><span style=\"display: inline-block; position: relative; width: 2.192em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1002.19em, 4.243em, -999.998em); top: -5.356em; left: 50%; margin-left: -1.116em;\"><span class=\"mtd\" id=\"MathJax-Span-1158\"><span class=\"mrow\" id=\"MathJax-Span-1159\"><span class=\"mi\" id=\"MathJax-Span-1160\" style=\"font-family: MathJax_Math-italic;\">a<\/span><span class=\"mo\" id=\"MathJax-Span-1161\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mi\" id=\"MathJax-Span-1162\" style=\"font-family: MathJax_Math-italic; padding-left: 0.235em;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.264em, 1002.19em, 4.243em, -999.998em); top: -3.959em; left: 50%; margin-left: -1.116em;\"><span class=\"mtd\" id=\"MathJax-Span-1173\"><span class=\"mrow\" id=\"MathJax-Span-1174\"><span class=\"mi\" id=\"MathJax-Span-1175\" style=\"font-family: MathJax_Math-italic;\">c<\/span><span class=\"mo\" id=\"MathJax-Span-1176\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mi\" id=\"MathJax-Span-1177\" style=\"font-family: MathJax_Math-italic; padding-left: 0.235em;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.42em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-1184\"><span class=\"mrow\" id=\"MathJax-Span-1185\"><span class=\"mi\" id=\"MathJax-Span-1186\" style=\"font-family: MathJax_Math-italic;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.522em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-1187\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-40\">\\left|\\begin{array}{ccc}  b+c & c+a & a+b \\\\  a+b & b+c & c+a \\\\  a & b & c  \\end{array}\\right|<\/script> = a<sup>2<\/sup> + b<sup>2<\/sup> + c<sup>2<\/sup> \u2013 3abc (Mar. 2008; May 2007)<\/h3>\n\n\n\n<p>Answer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-5969\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3d-2.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(d) 2\" width=\"295\" height=\"466\" sizes=\"auto, (max-width: 295px) 100vw, 295px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(d) 2\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(d)\"><br> = (a + b + c) [(c \u2013 b) (c \u2013 a) \u2013 (a \u2013 b) (b \u2013 a)]<br> = (a + b + c) [c<sup>2<\/sup> \u2013 bc \u2013 ac + ab + a<sup>2<\/sup> \u2013 2ab + b<sup>2<\/sup>]<br> = (a + b + c) [a<sup>2<\/sup> + b<sup>2<\/sup> + c<sup>2<\/sup> \u2013 ab \u2013 bc \u2013 ca]<br> = a<sup>2<\/sup> + b<sup>2<\/sup> + c<sup>2<\/sup> \u2013 3abc<\/p>\n\n\n\n<p>Question 3.<br> Show that <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-41-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-1188\" style=\"width: 11.279em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 9.695em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1009.55em, 6.247em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-1189\"><span class=\"mrow\" id=\"MathJax-Span-1190\"><span class=\"mo\" id=\"MathJax-Span-1191\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-1192\"><span style=\"display: inline-block; position: relative; width: 8.763em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.286em, 1002.19em, 5.967em, -999.998em); top: -4.378em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 2.239em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.264em, 1002.19em, 4.336em, -999.998em); top: -5.356em; left: 50%; margin-left: -1.116em;\"><span class=\"mtd\" id=\"MathJax-Span-1193\"><span class=\"mrow\" id=\"MathJax-Span-1194\"><span class=\"texatom\" id=\"MathJax-Span-1195\"><span class=\"mrow\" id=\"MathJax-Span-1196\"><span class=\"mi\" id=\"MathJax-Span-1197\" style=\"font-family: MathJax_Main;\">y<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-1198\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"texatom\" id=\"MathJax-Span-1199\" style=\"padding-left: 0.235em;\"><span class=\"mrow\" id=\"MathJax-Span-1200\"><span class=\"mi\" id=\"MathJax-Span-1201\" style=\"font-family: MathJax_Main;\">z<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.336em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.277em;\"><span class=\"mtd\" id=\"MathJax-Span-1212\"><span class=\"mrow\" id=\"MathJax-Span-1213\"><span class=\"texatom\" id=\"MathJax-Span-1214\"><span class=\"mrow\" id=\"MathJax-Span-1215\"><span class=\"mi\" id=\"MathJax-Span-1216\" style=\"font-family: MathJax_Main;\">y<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.42em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-1231\"><span class=\"mrow\" id=\"MathJax-Span-1232\"><span class=\"texatom\" id=\"MathJax-Span-1233\"><span class=\"mrow\" id=\"MathJax-Span-1234\"><span class=\"mi\" id=\"MathJax-Span-1235\" style=\"font-family: MathJax_Main;\">z<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.383em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1002.19em, 5.827em, -999.998em); top: -4.238em; left: 3.218em;\"><span style=\"display: inline-block; position: relative; width: 2.239em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.277em;\"><span class=\"mtd\" id=\"MathJax-Span-1202\"><span class=\"mrow\" id=\"MathJax-Span-1203\"><span class=\"texatom\" id=\"MathJax-Span-1204\"><span class=\"mrow\" id=\"MathJax-Span-1205\"><span class=\"mi\" id=\"MathJax-Span-1206\" style=\"font-family: MathJax_Main;\">x<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.264em, 1002.19em, 4.243em, -999.998em); top: -3.959em; left: 50%; margin-left: -1.116em;\"><span class=\"mtd\" id=\"MathJax-Span-1217\"><span class=\"mrow\" id=\"MathJax-Span-1218\"><span class=\"texatom\" id=\"MathJax-Span-1219\"><span class=\"mrow\" id=\"MathJax-Span-1220\"><span class=\"mi\" id=\"MathJax-Span-1221\" style=\"font-family: MathJax_Main;\">z<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-1222\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"texatom\" id=\"MathJax-Span-1223\" style=\"padding-left: 0.235em;\"><span class=\"mrow\" id=\"MathJax-Span-1224\"><span class=\"mi\" id=\"MathJax-Span-1225\" style=\"font-family: MathJax_Main;\">x<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.42em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-1236\"><span class=\"mrow\" id=\"MathJax-Span-1237\"><span class=\"texatom\" id=\"MathJax-Span-1238\"><span class=\"mrow\" id=\"MathJax-Span-1239\"><span class=\"mi\" id=\"MathJax-Span-1240\" style=\"font-family: MathJax_Main;\">z<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.243em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.519em, 1002.29em, 6.247em, -999.998em); top: -4.425em; left: 6.433em;\"><span style=\"display: inline-block; position: relative; width: 2.286em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.277em;\"><span class=\"mtd\" id=\"MathJax-Span-1207\"><span class=\"mrow\" id=\"MathJax-Span-1208\"><span class=\"texatom\" id=\"MathJax-Span-1209\"><span class=\"mrow\" id=\"MathJax-Span-1210\"><span class=\"mi\" id=\"MathJax-Span-1211\" style=\"font-family: MathJax_Main;\">x<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.336em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.277em;\"><span class=\"mtd\" id=\"MathJax-Span-1226\"><span class=\"mrow\" id=\"MathJax-Span-1227\"><span class=\"texatom\" id=\"MathJax-Span-1228\"><span class=\"mrow\" id=\"MathJax-Span-1229\"><span class=\"mi\" id=\"MathJax-Span-1230\" style=\"font-family: MathJax_Main;\">y<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.264em, 1002.29em, 4.336em, -999.998em); top: -2.561em; left: 50%; margin-left: -1.163em;\"><span class=\"mtd\" id=\"MathJax-Span-1241\"><span class=\"mrow\" id=\"MathJax-Span-1242\"><span class=\"texatom\" id=\"MathJax-Span-1243\"><span class=\"mrow\" id=\"MathJax-Span-1244\"><span class=\"mi\" id=\"MathJax-Span-1245\" style=\"font-family: MathJax_Main;\">x<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-1246\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"texatom\" id=\"MathJax-Span-1247\" style=\"padding-left: 0.235em;\"><span class=\"mrow\" id=\"MathJax-Span-1248\"><span class=\"mi\" id=\"MathJax-Span-1249\" style=\"font-family: MathJax_Main;\">y<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-1250\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.997em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-41\">\\left|\\begin{array}{ccc}  \\mathrm{y}+\\mathrm{z} & \\mathrm{x} & \\mathrm{x} \\\\  \\mathrm{y} & \\mathrm{z}+\\mathrm{x} & \\mathrm{y} \\\\  \\mathrm{z} & \\mathrm{z} & \\mathrm{x}+\\mathrm{y}  \\end{array}\\right|<\/script> = 4xyz.<br> Answer:<br> R<sub>1<\/sub> \u2013 (R<sub>2<\/sub> + R<sub>3<\/sub>) on the given determinant gives<br>\n\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-5968\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3d-3.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(d) 3\" width=\"225\" height=\"276\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(d) 3\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(d)\"><br> = 2[z(xy) \u2013 y(-xz)]<br> = 2[2xyz] = 4xyz = RHS<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Question 4.<br> If <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-42-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-1251\" style=\"width: 8.297em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 7.132em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1006.99em, 6.293em, -999.998em); top: -4.471em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-1252\"><span class=\"mrow\" id=\"MathJax-Span-1253\"><span class=\"mo\" id=\"MathJax-Span-1254\" style=\"vertical-align: 2.192em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.37em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.514em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.815em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.069em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-1255\"><span style=\"display: inline-block; position: relative; width: 6.2em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.379em, 1000.51em, 5.967em, -999.998em); top: -4.331em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.277em;\"><span class=\"mtd\" id=\"MathJax-Span-1256\"><span class=\"mrow\" id=\"MathJax-Span-1257\"><span class=\"mi\" id=\"MathJax-Span-1258\" style=\"font-family: MathJax_Math-italic;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.42em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-1271\"><span class=\"mrow\" id=\"MathJax-Span-1272\"><span class=\"mi\" id=\"MathJax-Span-1273\" style=\"font-family: MathJax_Math-italic;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.42em, 4.15em, -999.998em); top: -2.514em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-1286\"><span class=\"mrow\" id=\"MathJax-Span-1287\"><span class=\"mi\" id=\"MathJax-Span-1288\" style=\"font-family: MathJax_Math-italic;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.336em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.379em, 1000.98em, 6.34em, -999.998em); top: -4.704em; left: 1.54em;\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.031em, 1000.98em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.464em;\"><span class=\"mtd\" id=\"MathJax-Span-1259\"><span class=\"mrow\" id=\"MathJax-Span-1260\"><span class=\"msubsup\" id=\"MathJax-Span-1261\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-1262\" style=\"font-family: MathJax_Math-italic;\">a<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-1263\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.031em, 1000.84em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.417em;\"><span class=\"mtd\" id=\"MathJax-Span-1274\"><span class=\"mrow\" id=\"MathJax-Span-1275\"><span class=\"msubsup\" id=\"MathJax-Span-1276\"><span style=\"display: inline-block; position: relative; width: 0.841em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-1277\" style=\"font-family: MathJax_Math-italic;\">b<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.422em;\"><span class=\"mn\" id=\"MathJax-Span-1278\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.031em, 1000.84em, 4.15em, -999.998em); top: -2.514em; left: 50%; margin-left: -0.417em;\"><span class=\"mtd\" id=\"MathJax-Span-1289\"><span class=\"mrow\" id=\"MathJax-Span-1290\"><span class=\"msubsup\" id=\"MathJax-Span-1291\"><span style=\"display: inline-block; position: relative; width: 0.841em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-1292\" style=\"font-family: MathJax_Math-italic;\">c<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.422em;\"><span class=\"mn\" id=\"MathJax-Span-1293\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.709em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.472em, 1002.71em, 6.526em, -999.998em); top: -4.797em; left: 3.497em;\"><span style=\"display: inline-block; position: relative; width: 2.705em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.031em, 1002.71em, 4.243em, -999.998em); top: -5.356em; left: 50%; margin-left: -1.349em;\"><span class=\"mtd\" id=\"MathJax-Span-1264\"><span class=\"mrow\" id=\"MathJax-Span-1265\"><span class=\"mn\" id=\"MathJax-Span-1266\" style=\"font-family: MathJax_Main;\">1<\/span><span class=\"mo\" id=\"MathJax-Span-1267\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"msubsup\" id=\"MathJax-Span-1268\" style=\"padding-left: 0.235em;\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-1269\" style=\"font-family: MathJax_Math-italic;\">a<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-1270\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.031em, 1002.61em, 4.243em, -999.998em); top: -3.959em; left: 50%; margin-left: -1.302em;\"><span class=\"mtd\" id=\"MathJax-Span-1279\"><span class=\"mrow\" id=\"MathJax-Span-1280\"><span class=\"mn\" id=\"MathJax-Span-1281\" style=\"font-family: MathJax_Main;\">1<\/span><span class=\"mo\" id=\"MathJax-Span-1282\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"msubsup\" id=\"MathJax-Span-1283\" style=\"padding-left: 0.235em;\"><span style=\"display: inline-block; position: relative; width: 0.841em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-1284\" style=\"font-family: MathJax_Math-italic;\">b<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.422em;\"><span class=\"mn\" id=\"MathJax-Span-1285\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.031em, 1002.61em, 4.243em, -999.998em); top: -2.514em; left: 50%; margin-left: -1.302em;\"><span class=\"mtd\" id=\"MathJax-Span-1294\"><span class=\"mrow\" id=\"MathJax-Span-1295\"><span class=\"mn\" id=\"MathJax-Span-1296\" style=\"font-family: MathJax_Main;\">1<\/span><span class=\"mo\" id=\"MathJax-Span-1297\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"msubsup\" id=\"MathJax-Span-1298\" style=\"padding-left: 0.235em;\"><span style=\"display: inline-block; position: relative; width: 0.841em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-1299\" style=\"font-family: MathJax_Math-italic;\">c<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.422em;\"><span class=\"mn\" id=\"MathJax-Span-1300\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.802em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-1301\" style=\"vertical-align: 2.192em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.37em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.514em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.815em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.069em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.997em; border-left: 0px solid; width: 0px; height: 4.651em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-42\">\\left|\\begin{array}{ccc}  a & a^2 & 1+a^3 \\\\  b & b^2 & 1+b^3 \\\\  c & c^2 & 1+c^3  \\end{array}\\right|<\/script> = 0 and <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-43-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-1302\" style=\"width: 5.781em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.988em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1004.85em, 6.386em, -999.998em); top: -4.518em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-1303\"><span class=\"mrow\" id=\"MathJax-Span-1304\"><span class=\"mo\" id=\"MathJax-Span-1305\" style=\"vertical-align: 2.239em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.277em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.514em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.768em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.023em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-1306\"><span style=\"display: inline-block; position: relative; width: 4.056em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.379em, 1000.51em, 6.06em, -999.998em); top: -4.378em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.562em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.51em, 4.15em, -999.998em); top: -5.403em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-1307\"><span class=\"mrow\" id=\"MathJax-Span-1308\"><span class=\"texatom\" id=\"MathJax-Span-1309\"><span class=\"mrow\" id=\"MathJax-Span-1310\"><span class=\"mi\" id=\"MathJax-Span-1311\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.51em, 4.15em, -999.998em); top: -3.912em; left: 50%; margin-left: -0.277em;\"><span class=\"mtd\" id=\"MathJax-Span-1322\"><span class=\"mrow\" id=\"MathJax-Span-1323\"><span class=\"texatom\" id=\"MathJax-Span-1324\"><span class=\"mrow\" id=\"MathJax-Span-1325\"><span class=\"mi\" id=\"MathJax-Span-1326\" style=\"font-family: MathJax_Main;\">b<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.42em, 4.15em, -999.998em); top: -2.467em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-1337\"><span class=\"mrow\" id=\"MathJax-Span-1338\"><span class=\"texatom\" id=\"MathJax-Span-1339\"><span class=\"mrow\" id=\"MathJax-Span-1340\"><span class=\"mi\" id=\"MathJax-Span-1341\" style=\"font-family: MathJax_Main;\">c<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.383em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.379em, 1000.98em, 6.433em, -999.998em); top: -4.751em; left: 1.54em;\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.031em, 1000.93em, 4.15em, -999.998em); top: -5.403em; left: 50%; margin-left: -0.464em;\"><span class=\"mtd\" id=\"MathJax-Span-1312\"><span class=\"mrow\" id=\"MathJax-Span-1313\"><span class=\"msubsup\" id=\"MathJax-Span-1314\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-1315\"><span class=\"mrow\" id=\"MathJax-Span-1316\"><span class=\"mi\" id=\"MathJax-Span-1317\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-1318\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.985em, 1000.98em, 4.15em, -999.998em); top: -3.912em; left: 50%; margin-left: -0.51em;\"><span class=\"mtd\" id=\"MathJax-Span-1327\"><span class=\"mrow\" id=\"MathJax-Span-1328\"><span class=\"msubsup\" id=\"MathJax-Span-1329\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-1330\"><span class=\"mrow\" id=\"MathJax-Span-1331\"><span class=\"mi\" id=\"MathJax-Span-1332\" style=\"font-family: MathJax_Main;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.425em; left: 0.562em;\"><span class=\"mn\" id=\"MathJax-Span-1333\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.031em, 1000.89em, 4.15em, -999.998em); top: -2.467em; left: 50%; margin-left: -0.417em;\"><span class=\"mtd\" id=\"MathJax-Span-1342\"><span class=\"mrow\" id=\"MathJax-Span-1343\"><span class=\"msubsup\" id=\"MathJax-Span-1344\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-1345\"><span class=\"mrow\" id=\"MathJax-Span-1346\"><span class=\"mi\" id=\"MathJax-Span-1347\" style=\"font-family: MathJax_Main;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.468em;\"><span class=\"mn\" id=\"MathJax-Span-1348\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.755em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.425em, 1000.42em, 6.293em, -999.998em); top: -4.611em; left: 3.544em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.403em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-1319\"><span class=\"mrow\" id=\"MathJax-Span-1320\"><span class=\"mn\" id=\"MathJax-Span-1321\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.912em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-1334\"><span class=\"mrow\" id=\"MathJax-Span-1335\"><span class=\"mn\" id=\"MathJax-Span-1336\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.467em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-1349\"><span class=\"mrow\" id=\"MathJax-Span-1350\"><span class=\"mn\" id=\"MathJax-Span-1351\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.616em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-1352\" style=\"vertical-align: 2.239em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.277em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.514em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.768em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.023em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.522em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -2.051em; border-left: 0px solid; width: 0px; height: 4.705em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-43\">\\left|\\begin{array}{ccc}  \\mathrm{a} & \\mathrm{a}^2 & 1 \\\\  \\mathrm{b} & \\mathrm{b}^2 & 1 \\\\  \\mathrm{c} & \\mathrm{c}^2 & 1  \\end{array}\\right|<\/script> \u2260 0, then show that abc = -1. (Mar. \u201914)<\/h3>\n\n\n\n<p>Answer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-5967\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3d-4.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(d) 4\" width=\"332\" height=\"540\" sizes=\"auto, (max-width: 332px) 100vw, 332px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(d) 4\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(d)\"><br> \u21d2 abc + 1 = 0<br> \u21d2 abc = -1<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Question 5.<br> Without expanding the determinant, prove that<br> (i) <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-44-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-1353\" style=\"width: 14.308em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 12.304em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.192em, 1012.16em, 6.433em, -999.998em); top: -4.564em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-1354\"><span class=\"mrow\" id=\"MathJax-Span-1355\"><span class=\"mo\" id=\"MathJax-Span-1356\" style=\"vertical-align: 2.239em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.277em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.514em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.768em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.023em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-1357\"><span style=\"display: inline-block; position: relative; width: 4.616em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.379em, 1000.51em, 6.06em, -999.998em); top: -4.378em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.562em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.51em, 4.15em, -999.998em); top: -5.403em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1358\"><span class=\"mrow\" id=\"MathJax-Span-1359\"><span class=\"texatom\" id=\"MathJax-Span-1360\"><span class=\"mrow\" id=\"MathJax-Span-1361\"><span class=\"mi\" id=\"MathJax-Span-1362\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.51em, 4.15em, -999.998em); top: -3.912em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1376\"><span class=\"mrow\" id=\"MathJax-Span-1377\"><span class=\"texatom\" id=\"MathJax-Span-1378\"><span class=\"mrow\" id=\"MathJax-Span-1379\"><span class=\"mi\" id=\"MathJax-Span-1380\" style=\"font-family: MathJax_Main;\">b<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.42em, 4.15em, -999.998em); top: -2.467em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1394\"><span class=\"mrow\" id=\"MathJax-Span-1395\"><span class=\"texatom\" id=\"MathJax-Span-1396\"><span class=\"mrow\" id=\"MathJax-Span-1397\"><span class=\"mi\" id=\"MathJax-Span-1398\" style=\"font-family: MathJax_Main;\">c<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.383em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.379em, 1000.98em, 6.433em, -999.998em); top: -4.751em; left: 1.54em;\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.031em, 1000.93em, 4.15em, -999.998em); top: -5.403em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1363\"><span class=\"mrow\" id=\"MathJax-Span-1364\"><span class=\"msubsup\" id=\"MathJax-Span-1365\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-1366\"><span class=\"mrow\" id=\"MathJax-Span-1367\"><span class=\"mi\" id=\"MathJax-Span-1368\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-1369\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.985em, 1000.98em, 4.15em, -999.998em); top: -3.912em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1381\"><span class=\"mrow\" id=\"MathJax-Span-1382\"><span class=\"msubsup\" id=\"MathJax-Span-1383\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-1384\"><span class=\"mrow\" id=\"MathJax-Span-1385\"><span class=\"mi\" id=\"MathJax-Span-1386\" style=\"font-family: MathJax_Main;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.425em; left: 0.562em;\"><span class=\"mn\" id=\"MathJax-Span-1387\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.031em, 1000.89em, 4.15em, -999.998em); top: -2.467em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1399\"><span class=\"mrow\" id=\"MathJax-Span-1400\"><span class=\"msubsup\" id=\"MathJax-Span-1401\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-1402\"><span class=\"mrow\" id=\"MathJax-Span-1403\"><span class=\"mi\" id=\"MathJax-Span-1404\" style=\"font-family: MathJax_Main;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.468em;\"><span class=\"mn\" id=\"MathJax-Span-1405\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.755em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.379em, 1001.03em, 6.293em, -999.998em); top: -4.611em; left: 3.544em;\"><span style=\"display: inline-block; position: relative; width: 1.074em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.98em, 4.15em, -999.998em); top: -5.403em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1370\"><span class=\"mrow\" id=\"MathJax-Span-1371\"><span class=\"texatom\" id=\"MathJax-Span-1372\"><span class=\"mrow\" id=\"MathJax-Span-1373\"><span class=\"mi\" id=\"MathJax-Span-1374\" style=\"font-family: MathJax_Main;\">b<\/span><span class=\"mi\" id=\"MathJax-Span-1375\" style=\"font-family: MathJax_Main;\">c<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.93em, 4.15em, -999.998em); top: -3.912em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1388\"><span class=\"mrow\" id=\"MathJax-Span-1389\"><span class=\"texatom\" id=\"MathJax-Span-1390\"><span class=\"mrow\" id=\"MathJax-Span-1391\"><span class=\"mi\" id=\"MathJax-Span-1392\" style=\"font-family: MathJax_Main;\">c<\/span><span class=\"mi\" id=\"MathJax-Span-1393\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1001.03em, 4.15em, -999.998em); top: -2.467em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1406\"><span class=\"mrow\" id=\"MathJax-Span-1407\"><span class=\"texatom\" id=\"MathJax-Span-1408\"><span class=\"mrow\" id=\"MathJax-Span-1409\"><span class=\"mi\" id=\"MathJax-Span-1410\" style=\"font-family: MathJax_Main;\">a<\/span><span class=\"mi\" id=\"MathJax-Span-1411\" style=\"font-family: MathJax_Main;\">b<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.616em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-1412\" style=\"vertical-align: 2.239em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.277em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.514em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.768em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.023em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-1413\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"mrow\" id=\"MathJax-Span-1414\" style=\"padding-left: 0.282em;\"><span class=\"mo\" id=\"MathJax-Span-1415\" style=\"vertical-align: 2.239em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.277em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.514em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.768em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.023em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-1416\"><span style=\"display: inline-block; position: relative; width: 4.476em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.425em, 1000.42em, 6.293em, -999.998em); top: -4.611em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.403em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-1417\"><span class=\"mrow\" id=\"MathJax-Span-1418\"><span class=\"mn\" id=\"MathJax-Span-1419\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.912em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-1434\"><span class=\"mrow\" id=\"MathJax-Span-1435\"><span class=\"mn\" id=\"MathJax-Span-1436\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.467em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-1451\"><span class=\"mrow\" id=\"MathJax-Span-1452\"><span class=\"mn\" id=\"MathJax-Span-1453\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.616em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.379em, 1000.98em, 6.433em, -999.998em); top: -4.751em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.031em, 1000.93em, 4.15em, -999.998em); top: -5.403em; left: 50%; margin-left: -0.464em;\"><span class=\"mtd\" id=\"MathJax-Span-1420\"><span class=\"mrow\" id=\"MathJax-Span-1421\"><span class=\"msubsup\" id=\"MathJax-Span-1422\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-1423\"><span class=\"mrow\" id=\"MathJax-Span-1424\"><span class=\"mi\" id=\"MathJax-Span-1425\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-1426\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.985em, 1000.98em, 4.15em, -999.998em); top: -3.912em; left: 50%; margin-left: -0.51em;\"><span class=\"mtd\" id=\"MathJax-Span-1437\"><span class=\"mrow\" id=\"MathJax-Span-1438\"><span class=\"msubsup\" id=\"MathJax-Span-1439\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-1440\"><span class=\"mrow\" id=\"MathJax-Span-1441\"><span class=\"mi\" id=\"MathJax-Span-1442\" style=\"font-family: MathJax_Main;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.425em; left: 0.562em;\"><span class=\"mn\" id=\"MathJax-Span-1443\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.031em, 1000.89em, 4.15em, -999.998em); top: -2.467em; left: 50%; margin-left: -0.417em;\"><span class=\"mtd\" id=\"MathJax-Span-1454\"><span class=\"mrow\" id=\"MathJax-Span-1455\"><span class=\"msubsup\" id=\"MathJax-Span-1456\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-1457\"><span class=\"mrow\" id=\"MathJax-Span-1458\"><span class=\"mi\" id=\"MathJax-Span-1459\" style=\"font-family: MathJax_Main;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.468em;\"><span class=\"mn\" id=\"MathJax-Span-1460\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.755em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.379em, 1000.98em, 6.433em, -999.998em); top: -4.751em; left: 3.497em;\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.031em, 1000.93em, 4.15em, -999.998em); top: -5.403em; left: 50%; margin-left: -0.464em;\"><span class=\"mtd\" id=\"MathJax-Span-1427\"><span class=\"mrow\" id=\"MathJax-Span-1428\"><span class=\"msubsup\" id=\"MathJax-Span-1429\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-1430\"><span class=\"mrow\" id=\"MathJax-Span-1431\"><span class=\"mi\" id=\"MathJax-Span-1432\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-1433\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.985em, 1000.98em, 4.15em, -999.998em); top: -3.912em; left: 50%; margin-left: -0.51em;\"><span class=\"mtd\" id=\"MathJax-Span-1444\"><span class=\"mrow\" id=\"MathJax-Span-1445\"><span class=\"msubsup\" id=\"MathJax-Span-1446\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-1447\"><span class=\"mrow\" id=\"MathJax-Span-1448\"><span class=\"mi\" id=\"MathJax-Span-1449\" style=\"font-family: MathJax_Main;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.425em; left: 0.562em;\"><span class=\"mn\" id=\"MathJax-Span-1450\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.031em, 1000.89em, 4.15em, -999.998em); top: -2.467em; left: 50%; margin-left: -0.417em;\"><span class=\"mtd\" id=\"MathJax-Span-1461\"><span class=\"mrow\" id=\"MathJax-Span-1462\"><span class=\"msubsup\" id=\"MathJax-Span-1463\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-1464\"><span class=\"mrow\" id=\"MathJax-Span-1465\"><span class=\"mi\" id=\"MathJax-Span-1466\" style=\"font-family: MathJax_Main;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.468em;\"><span class=\"mn\" id=\"MathJax-Span-1467\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.755em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-1468\" style=\"vertical-align: 2.239em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.277em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.514em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.768em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.023em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -2.051em; border-left: 0px solid; width: 0px; height: 4.705em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-44\">\\left|\\begin{array}{lll}  \\mathrm{a} & \\mathrm{a}^2 & \\mathrm{b c} \\\\  \\mathrm{b} & \\mathrm{b}^2 & \\mathrm{c a} \\\\  \\mathrm{c} & \\mathrm{c}^2 & \\mathrm{a b}  \\end{array}\\right|=\\left|\\begin{array}{ccc}  1 & \\mathrm{a}^2 & \\mathrm{a}^3 \\\\  1 & \\mathrm{b}^2 & \\mathrm{b}^3 \\\\  1 & \\mathrm{c}^2 & \\mathrm{c}^3  \\end{array}\\right|<\/script><\/h3>\n\n\n\n<p>Answer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-5966\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3d-5.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(d) 5\" width=\"245\" height=\"369\" sizes=\"auto, (max-width: 245px) 100vw, 245px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(d) 5\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(d)\"><\/p>\n\n\n\n<p>(ii) <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-45-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-1469\" style=\"width: 15.333em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 13.19em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.192em, 1013.05em, 6.34em, -999.998em); top: -4.518em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-1470\"><span class=\"mrow\" id=\"MathJax-Span-1471\"><span class=\"mo\" id=\"MathJax-Span-1472\" style=\"vertical-align: 2.192em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.417em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.349em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-1473\"><span style=\"display: inline-block; position: relative; width: 4.988em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1001.03em, 5.92em, -999.998em); top: -4.285em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 1.027em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1001.03em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.51em;\"><span class=\"mtd\" id=\"MathJax-Span-1474\"><span class=\"mrow\" id=\"MathJax-Span-1475\"><span class=\"texatom\" id=\"MathJax-Span-1476\"><span class=\"mrow\" id=\"MathJax-Span-1477\"><span class=\"mi\" id=\"MathJax-Span-1478\" style=\"font-family: MathJax_Main;\">a<\/span><span class=\"mi\" id=\"MathJax-Span-1479\" style=\"font-family: MathJax_Main;\">x<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.031em, 1000.98em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.464em;\"><span class=\"mtd\" id=\"MathJax-Span-1492\"><span class=\"mrow\" id=\"MathJax-Span-1493\"><span class=\"msubsup\" id=\"MathJax-Span-1494\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-1495\"><span class=\"mrow\" id=\"MathJax-Span-1496\"><span class=\"mi\" id=\"MathJax-Span-1497\" style=\"font-family: MathJax_Main;\">x<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-1498\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.514em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-1513\"><span class=\"mrow\" id=\"MathJax-Span-1514\"><span class=\"mn\" id=\"MathJax-Span-1515\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.289em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1001.07em, 6.153em, -999.998em); top: -4.518em; left: 2.053em;\"><span style=\"display: inline-block; position: relative; width: 1.074em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.07em, 4.336em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.557em;\"><span class=\"mtd\" id=\"MathJax-Span-1480\"><span class=\"mrow\" id=\"MathJax-Span-1481\"><span class=\"texatom\" id=\"MathJax-Span-1482\"><span class=\"mrow\" id=\"MathJax-Span-1483\"><span class=\"mi\" id=\"MathJax-Span-1484\" style=\"font-family: MathJax_Main;\">b<\/span><span class=\"mi\" id=\"MathJax-Span-1485\" style=\"font-family: MathJax_Main;\">y<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.031em, 1000.98em, 4.336em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.464em;\"><span class=\"mtd\" id=\"MathJax-Span-1499\"><span class=\"mrow\" id=\"MathJax-Span-1500\"><span class=\"msubsup\" id=\"MathJax-Span-1501\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.336em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-1502\"><span class=\"mrow\" id=\"MathJax-Span-1503\"><span class=\"mi\" id=\"MathJax-Span-1504\" style=\"font-family: MathJax_Main;\">y<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-1505\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.514em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-1516\"><span class=\"mrow\" id=\"MathJax-Span-1517\"><span class=\"mn\" id=\"MathJax-Span-1518\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.522em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.89em, 5.92em, -999.998em); top: -4.285em; left: 4.103em;\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.84em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.464em;\"><span class=\"mtd\" id=\"MathJax-Span-1486\"><span class=\"mrow\" id=\"MathJax-Span-1487\"><span class=\"texatom\" id=\"MathJax-Span-1488\"><span class=\"mrow\" id=\"MathJax-Span-1489\"><span class=\"mi\" id=\"MathJax-Span-1490\" style=\"font-family: MathJax_Main;\">c<\/span><span class=\"mi\" id=\"MathJax-Span-1491\" style=\"font-family: MathJax_Main;\">z<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.031em, 1000.89em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.417em;\"><span class=\"mtd\" id=\"MathJax-Span-1506\"><span class=\"mrow\" id=\"MathJax-Span-1507\"><span class=\"msubsup\" id=\"MathJax-Span-1508\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-1509\"><span class=\"mrow\" id=\"MathJax-Span-1510\"><span class=\"mi\" id=\"MathJax-Span-1511\" style=\"font-family: MathJax_Main;\">z<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.468em;\"><span class=\"mn\" id=\"MathJax-Span-1512\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.514em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-1519\"><span class=\"mrow\" id=\"MathJax-Span-1520\"><span class=\"mn\" id=\"MathJax-Span-1521\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.289em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-1522\" style=\"vertical-align: 2.192em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.417em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.349em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-1523\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"mrow\" id=\"MathJax-Span-1524\" style=\"padding-left: 0.282em;\"><span class=\"mo\" id=\"MathJax-Span-1525\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-1526\"><span style=\"display: inline-block; position: relative; width: 4.988em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.519em, 1000.93em, 6.247em, -999.998em); top: -4.471em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.51em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-1527\"><span class=\"mrow\" id=\"MathJax-Span-1528\"><span class=\"texatom\" id=\"MathJax-Span-1529\"><span class=\"mrow\" id=\"MathJax-Span-1530\"><span class=\"mi\" id=\"MathJax-Span-1531\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.277em;\"><span class=\"mtd\" id=\"MathJax-Span-1542\"><span class=\"mrow\" id=\"MathJax-Span-1543\"><span class=\"texatom\" id=\"MathJax-Span-1544\"><span class=\"mrow\" id=\"MathJax-Span-1545\"><span class=\"mi\" id=\"MathJax-Span-1546\" style=\"font-family: MathJax_Main;\">x<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.93em, 4.336em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.464em;\"><span class=\"mtd\" id=\"MathJax-Span-1557\"><span class=\"mrow\" id=\"MathJax-Span-1558\"><span class=\"texatom\" id=\"MathJax-Span-1559\"><span class=\"mrow\" id=\"MathJax-Span-1560\"><span class=\"mi\" id=\"MathJax-Span-1561\" style=\"font-family: MathJax_Main;\">y<\/span><span class=\"mi\" id=\"MathJax-Span-1562\" style=\"font-family: MathJax_Main;\">z<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.98em, 6.107em, -999.998em); top: -4.518em; left: 1.959em;\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.51em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.277em;\"><span class=\"mtd\" id=\"MathJax-Span-1532\"><span class=\"mrow\" id=\"MathJax-Span-1533\"><span class=\"texatom\" id=\"MathJax-Span-1534\"><span class=\"mrow\" id=\"MathJax-Span-1535\"><span class=\"mi\" id=\"MathJax-Span-1536\" style=\"font-family: MathJax_Main;\">b<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.336em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.277em;\"><span class=\"mtd\" id=\"MathJax-Span-1547\"><span class=\"mrow\" id=\"MathJax-Span-1548\"><span class=\"texatom\" id=\"MathJax-Span-1549\"><span class=\"mrow\" id=\"MathJax-Span-1550\"><span class=\"mi\" id=\"MathJax-Span-1551\" style=\"font-family: MathJax_Main;\">y<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.98em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.464em;\"><span class=\"mtd\" id=\"MathJax-Span-1563\"><span class=\"mrow\" id=\"MathJax-Span-1564\"><span class=\"texatom\" id=\"MathJax-Span-1565\"><span class=\"mrow\" id=\"MathJax-Span-1566\"><span class=\"mi\" id=\"MathJax-Span-1567\" style=\"font-family: MathJax_Main;\">z<\/span><span class=\"mi\" id=\"MathJax-Span-1568\" style=\"font-family: MathJax_Main;\">x<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.522em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.519em, 1001.03em, 6.247em, -999.998em); top: -4.471em; left: 3.963em;\"><span style=\"display: inline-block; position: relative; width: 1.074em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.42em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-1537\"><span class=\"mrow\" id=\"MathJax-Span-1538\"><span class=\"texatom\" id=\"MathJax-Span-1539\"><span class=\"mrow\" id=\"MathJax-Span-1540\"><span class=\"mi\" id=\"MathJax-Span-1541\" style=\"font-family: MathJax_Main;\">c<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.42em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-1552\"><span class=\"mrow\" id=\"MathJax-Span-1553\"><span class=\"texatom\" id=\"MathJax-Span-1554\"><span class=\"mrow\" id=\"MathJax-Span-1555\"><span class=\"mi\" id=\"MathJax-Span-1556\" style=\"font-family: MathJax_Main;\">z<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1001.03em, 4.336em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.51em;\"><span class=\"mtd\" id=\"MathJax-Span-1569\"><span class=\"mrow\" id=\"MathJax-Span-1570\"><span class=\"texatom\" id=\"MathJax-Span-1571\"><span class=\"mrow\" id=\"MathJax-Span-1572\"><span class=\"mi\" id=\"MathJax-Span-1573\" style=\"font-family: MathJax_Main;\">x<\/span><span class=\"mi\" id=\"MathJax-Span-1574\" style=\"font-family: MathJax_Main;\">y<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-1575\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.522em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.997em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-45\">\\left|\\begin{array}{ccc}  \\mathrm{a x} & \\mathrm{b y} & \\mathrm{c z} \\\\  \\mathrm{x}^2 & \\mathrm{y}^2 & \\mathrm{z}^2 \\\\  1 & 1 & 1  \\end{array}\\right|=\\left|\\begin{array}{ccc}  \\mathrm{a} & \\mathrm{b} & \\mathrm{c} \\\\  \\mathrm{x} & \\mathrm{y} & \\mathrm{z} \\\\  \\mathrm{y z} & \\mathrm{z x} & \\mathrm{x y}  \\end{array}\\right|<\/script><br>\n\nAnswer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-5965\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3d-6.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(d) 6\" width=\"244\" height=\"439\" sizes=\"auto, (max-width: 244px) 100vw, 244px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(d) 6\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(d)\"><\/p>\n\n\n\n<p>(iii) <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-46-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-1576\" style=\"width: 15.007em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 12.91em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.192em, 1012.77em, 6.34em, -999.998em); top: -4.518em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-1577\"><span class=\"mrow\" id=\"MathJax-Span-1578\"><span class=\"mo\" id=\"MathJax-Span-1579\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-1580\"><span style=\"display: inline-block; position: relative; width: 5.687em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.42em, 6.06em, -999.998em); top: -4.471em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1581\"><span class=\"mrow\" id=\"MathJax-Span-1582\"><span class=\"mn\" id=\"MathJax-Span-1583\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1593\"><span class=\"mrow\" id=\"MathJax-Span-1594\"><span class=\"mn\" id=\"MathJax-Span-1595\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1605\"><span class=\"mrow\" id=\"MathJax-Span-1606\"><span class=\"mn\" id=\"MathJax-Span-1607\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.93em, 6.107em, -999.998em); top: -4.518em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.84em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1584\"><span class=\"mrow\" id=\"MathJax-Span-1585\"><span class=\"mi\" id=\"MathJax-Span-1586\" style=\"font-family: MathJax_Math-italic;\">b<\/span><span class=\"mi\" id=\"MathJax-Span-1587\" style=\"font-family: MathJax_Math-italic;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.93em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1596\"><span class=\"mrow\" id=\"MathJax-Span-1597\"><span class=\"mi\" id=\"MathJax-Span-1598\" style=\"font-family: MathJax_Math-italic;\">c<\/span><span class=\"mi\" id=\"MathJax-Span-1599\" style=\"font-family: MathJax_Math-italic;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.93em, 4.15em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1608\"><span class=\"mrow\" id=\"MathJax-Span-1609\"><span class=\"mi\" id=\"MathJax-Span-1610\" style=\"font-family: MathJax_Math-italic;\">a<\/span><span class=\"mi\" id=\"MathJax-Span-1611\" style=\"font-family: MathJax_Math-italic;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.522em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.379em, 1002.19em, 6.247em, -999.998em); top: -4.564em; left: 3.451em;\"><span style=\"display: inline-block; position: relative; width: 2.192em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1002.1em, 4.243em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1588\"><span class=\"mrow\" id=\"MathJax-Span-1589\"><span class=\"mi\" id=\"MathJax-Span-1590\" style=\"font-family: MathJax_Math-italic;\">b<\/span><span class=\"mo\" id=\"MathJax-Span-1591\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mi\" id=\"MathJax-Span-1592\" style=\"font-family: MathJax_Math-italic; padding-left: 0.235em;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.264em, 1002.19em, 4.243em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1600\"><span class=\"mrow\" id=\"MathJax-Span-1601\"><span class=\"mi\" id=\"MathJax-Span-1602\" style=\"font-family: MathJax_Math-italic;\">c<\/span><span class=\"mo\" id=\"MathJax-Span-1603\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mi\" id=\"MathJax-Span-1604\" style=\"font-family: MathJax_Math-italic; padding-left: 0.235em;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1002.19em, 4.243em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1612\"><span class=\"mrow\" id=\"MathJax-Span-1613\"><span class=\"mi\" id=\"MathJax-Span-1614\" style=\"font-family: MathJax_Math-italic;\">a<\/span><span class=\"mo\" id=\"MathJax-Span-1615\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mi\" id=\"MathJax-Span-1616\" style=\"font-family: MathJax_Math-italic; padding-left: 0.235em;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-1617\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-1618\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"mrow\" id=\"MathJax-Span-1619\" style=\"padding-left: 0.282em;\"><span class=\"mo\" id=\"MathJax-Span-1620\" style=\"vertical-align: 2.192em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.37em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.514em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.815em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.069em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-1621\"><span style=\"display: inline-block; position: relative; width: 4.01em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.42em, 6.153em, -999.998em); top: -4.518em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1622\"><span class=\"mrow\" id=\"MathJax-Span-1623\"><span class=\"mn\" id=\"MathJax-Span-1624\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1633\"><span class=\"mrow\" id=\"MathJax-Span-1634\"><span class=\"mn\" id=\"MathJax-Span-1635\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.514em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1644\"><span class=\"mrow\" id=\"MathJax-Span-1645\"><span class=\"mn\" id=\"MathJax-Span-1646\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.522em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.379em, 1000.51em, 5.967em, -999.998em); top: -4.331em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1625\"><span class=\"mrow\" id=\"MathJax-Span-1626\"><span class=\"mi\" id=\"MathJax-Span-1627\" style=\"font-family: MathJax_Math-italic;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.42em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1636\"><span class=\"mrow\" id=\"MathJax-Span-1637\"><span class=\"mi\" id=\"MathJax-Span-1638\" style=\"font-family: MathJax_Math-italic;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.42em, 4.15em, -999.998em); top: -2.514em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1647\"><span class=\"mrow\" id=\"MathJax-Span-1648\"><span class=\"mi\" id=\"MathJax-Span-1649\" style=\"font-family: MathJax_Math-italic;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.336em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.379em, 1000.98em, 6.34em, -999.998em); top: -4.704em; left: 3.031em;\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.031em, 1000.98em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1628\"><span class=\"mrow\" id=\"MathJax-Span-1629\"><span class=\"msubsup\" id=\"MathJax-Span-1630\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-1631\" style=\"font-family: MathJax_Math-italic;\">a<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-1632\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.031em, 1000.84em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1639\"><span class=\"mrow\" id=\"MathJax-Span-1640\"><span class=\"msubsup\" id=\"MathJax-Span-1641\"><span style=\"display: inline-block; position: relative; width: 0.841em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-1642\" style=\"font-family: MathJax_Math-italic;\">b<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.422em;\"><span class=\"mn\" id=\"MathJax-Span-1643\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.031em, 1000.84em, 4.15em, -999.998em); top: -2.514em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1650\"><span class=\"mrow\" id=\"MathJax-Span-1651\"><span class=\"msubsup\" id=\"MathJax-Span-1652\"><span style=\"display: inline-block; position: relative; width: 0.841em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-1653\" style=\"font-family: MathJax_Math-italic;\">c<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.422em;\"><span class=\"mn\" id=\"MathJax-Span-1654\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.709em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-1655\" style=\"vertical-align: 2.192em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.37em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.514em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.815em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.069em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.522em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.997em; border-left: 0px solid; width: 0px; height: 4.651em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-46\">\\left|\\begin{array}{lll}  1 & b c & b+c \\\\  1 & c a & c+a \\\\  1 & a b & a+b  \\end{array}\\right|=\\left|\\begin{array}{lll}  1 & a & a^2 \\\\  1 & b & b^2 \\\\  1 & c & c^2  \\end{array}\\right|<\/script> (Board Model Paper)<br>\n\nAnswer:<br>\n\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-5964\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3d-7.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(d) 7\" width=\"325\" height=\"407\" sizes=\"auto, (max-width: 325px) 100vw, 325px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(d) 7\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(d)\"><br> (\u2235 R<sub>2<\/sub> \u2013 R<sub>1<\/sub>; R<sub>3<\/sub> \u2013 R<sub>1<\/sub>)<br> = (b \u2013 a) (c<sup>2<\/sup> \u2013 a<sup>2<\/sup>) \u2013 (c \u2013 a) (b<sup>2<\/sup> \u2013 a<sup>2<\/sup>)<br> = (b \u2013 a) (c \u2013 a) (c + a) \u2013 (c \u2013 a) (b \u2013 a) (b + a)<br> = (b \u2013 a) (c \u2013 a) (c + a \u2013 b \u2013 a)<br> = (b \u2013 a) (c \u2013 a) (c \u2013 b)<br> = (a \u2013 b) (b \u2013 c) (c \u2013 a)<br> LHS = RHS<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Question 6.<br> If \u0394<sub>1<\/sub> = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-47-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-1656\" style=\"width: 23.907em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 20.599em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1020.46em, 6.619em, -999.998em); top: -4.611em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-1657\"><span class=\"mrow\" id=\"MathJax-Span-1658\"><span class=\"mo\" id=\"MathJax-Span-1659\" style=\"vertical-align: 2.332em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.044em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.467em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.675em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -0.836em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-1660\"><span style=\"display: inline-block; position: relative; width: 19.667em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.565em, 1005.27em, 6.852em, -999.998em); top: -5.03em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 5.268em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.078em, 1005.27em, 4.476em, -999.998em); top: -5.543em; left: 50%; margin-left: -2.654em;\"><span class=\"mtd\" id=\"MathJax-Span-1661\"><span class=\"mrow\" id=\"MathJax-Span-1662\"><span class=\"msubsup\" id=\"MathJax-Span-1663\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-1664\"><span class=\"mrow\" id=\"MathJax-Span-1665\"><span class=\"mi\" id=\"MathJax-Span-1666\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.42em, 4.15em, -999.998em); top: -4.331em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-1667\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.42em, 4.15em, -999.998em); top: -3.679em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-1668\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-1669\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"msubsup\" id=\"MathJax-Span-1670\" style=\"padding-left: 0.235em;\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-1671\"><span class=\"mrow\" id=\"MathJax-Span-1672\"><span class=\"mi\" id=\"MathJax-Span-1673\" style=\"font-family: MathJax_Main;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.865em; left: 0.562em;\"><span class=\"mn\" id=\"MathJax-Span-1674\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-1675\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"msubsup\" id=\"MathJax-Span-1676\" style=\"padding-left: 0.235em;\"><span style=\"display: inline-block; position: relative; width: 0.841em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-1677\" style=\"font-family: MathJax_Math-italic;\">c<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.865em; left: 0.422em;\"><span class=\"mn\" id=\"MathJax-Span-1678\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1004.1em, 4.289em, -999.998em); top: -3.959em; left: 50%; margin-left: -2.048em;\"><span class=\"mtd\" id=\"MathJax-Span-1723\"><span class=\"mrow\" id=\"MathJax-Span-1724\"><span class=\"msubsup\" id=\"MathJax-Span-1725\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-1726\"><span class=\"mrow\" id=\"MathJax-Span-1727\"><span class=\"mi\" id=\"MathJax-Span-1728\" style=\"font-family: MathJax_Main;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.865em; left: 0.562em;\"><span class=\"mn\" id=\"MathJax-Span-1729\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"msubsup\" id=\"MathJax-Span-1730\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-1731\"><span class=\"mrow\" id=\"MathJax-Span-1732\"><span class=\"mi\" id=\"MathJax-Span-1733\" style=\"font-family: MathJax_Main;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.865em; left: 0.562em;\"><span class=\"mn\" id=\"MathJax-Span-1734\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-1735\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"msubsup\" id=\"MathJax-Span-1736\" style=\"padding-left: 0.235em;\"><span style=\"display: inline-block; position: relative; width: 0.841em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-1737\" style=\"font-family: MathJax_Math-italic;\">c<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.865em; left: 0.422em;\"><span class=\"mn\" id=\"MathJax-Span-1738\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1001.73em, 4.336em, -999.998em); top: -2.467em; left: 50%; margin-left: -0.883em;\"><span class=\"mtd\" id=\"MathJax-Span-1771\"><span class=\"mrow\" id=\"MathJax-Span-1772\"><span class=\"msubsup\" id=\"MathJax-Span-1773\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-1774\"><span class=\"mrow\" id=\"MathJax-Span-1775\"><span class=\"mi\" id=\"MathJax-Span-1776\" style=\"font-family: MathJax_Main;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.865em; left: 0.468em;\"><span class=\"mn\" id=\"MathJax-Span-1777\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"msubsup\" id=\"MathJax-Span-1778\"><span style=\"display: inline-block; position: relative; width: 0.841em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-1779\" style=\"font-family: MathJax_Math-italic;\">c<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.865em; left: 0.422em;\"><span class=\"mn\" id=\"MathJax-Span-1780\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 5.035em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1006.2em, 6.759em, -999.998em); top: -4.937em; left: 6.293em;\"><span style=\"display: inline-block; position: relative; width: 6.2em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1006.2em, 4.289em, -999.998em); top: -5.543em; left: 50%; margin-left: -3.12em;\"><span class=\"mtd\" id=\"MathJax-Span-1679\"><span class=\"mrow\" id=\"MathJax-Span-1680\"><span class=\"msubsup\" id=\"MathJax-Span-1681\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-1682\"><span class=\"mrow\" id=\"MathJax-Span-1683\"><span class=\"mi\" id=\"MathJax-Span-1684\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.865em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-1685\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"msubsup\" id=\"MathJax-Span-1686\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-1687\"><span class=\"mrow\" id=\"MathJax-Span-1688\"><span class=\"mi\" id=\"MathJax-Span-1689\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.865em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-1690\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-1691\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"msubsup\" id=\"MathJax-Span-1692\" style=\"padding-left: 0.235em;\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-1693\"><span class=\"mrow\" id=\"MathJax-Span-1694\"><span class=\"mi\" id=\"MathJax-Span-1695\" style=\"font-family: MathJax_Main;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.865em; left: 0.562em;\"><span class=\"mn\" id=\"MathJax-Span-1696\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-1697\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"msubsup\" id=\"MathJax-Span-1698\" style=\"padding-left: 0.235em;\"><span style=\"display: inline-block; position: relative; width: 0.841em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-1699\" style=\"font-family: MathJax_Math-italic;\">c<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.865em; left: 0.422em;\"><span class=\"mn\" id=\"MathJax-Span-1700\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.985em, 1003.12em, 4.383em, -999.998em); top: -3.959em; left: 50%; margin-left: -1.535em;\"><span class=\"mtd\" id=\"MathJax-Span-1739\"><span class=\"mrow\" id=\"MathJax-Span-1740\"><span class=\"msubsup\" id=\"MathJax-Span-1741\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-1742\"><span class=\"mrow\" id=\"MathJax-Span-1743\"><span class=\"mi\" id=\"MathJax-Span-1744\" style=\"font-family: MathJax_Main;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.42em, 4.15em, -999.998em); top: -4.425em; left: 0.562em;\"><span class=\"mn\" id=\"MathJax-Span-1745\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.42em, 4.15em, -999.998em); top: -3.772em; left: 0.562em;\"><span class=\"mn\" id=\"MathJax-Span-1746\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-1747\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"msubsup\" id=\"MathJax-Span-1748\" style=\"padding-left: 0.235em;\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-1749\"><span class=\"mrow\" id=\"MathJax-Span-1750\"><span class=\"mi\" id=\"MathJax-Span-1751\" style=\"font-family: MathJax_Main;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.865em; left: 0.468em;\"><span class=\"mn\" id=\"MathJax-Span-1752\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1001.73em, 4.336em, -999.998em); top: -2.467em; left: 50%; margin-left: -0.883em;\"><span class=\"mtd\" id=\"MathJax-Span-1781\"><span class=\"mrow\" id=\"MathJax-Span-1782\"><span class=\"msubsup\" id=\"MathJax-Span-1783\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-1784\"><span class=\"mrow\" id=\"MathJax-Span-1785\"><span class=\"mi\" id=\"MathJax-Span-1786\" style=\"font-family: MathJax_Main;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.865em; left: 0.468em;\"><span class=\"mn\" id=\"MathJax-Span-1787\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"msubsup\" id=\"MathJax-Span-1788\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-1789\"><span class=\"mrow\" id=\"MathJax-Span-1790\"><span class=\"mi\" id=\"MathJax-Span-1791\" style=\"font-family: MathJax_Main;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.865em; left: 0.468em;\"><span class=\"mn\" id=\"MathJax-Span-1792\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.942em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.705em, 1006.2em, 7.085em, -999.998em); top: -5.077em; left: 13.469em;\"><span style=\"display: inline-block; position: relative; width: 6.2em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1006.2em, 4.336em, -999.998em); top: -5.543em; left: 50%; margin-left: -3.12em;\"><span class=\"mtd\" id=\"MathJax-Span-1701\"><span class=\"mrow\" id=\"MathJax-Span-1702\"><span class=\"msubsup\" id=\"MathJax-Span-1703\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-1704\"><span class=\"mrow\" id=\"MathJax-Span-1705\"><span class=\"mi\" id=\"MathJax-Span-1706\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.865em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-1707\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"msubsup\" id=\"MathJax-Span-1708\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-1709\"><span class=\"mrow\" id=\"MathJax-Span-1710\"><span class=\"mi\" id=\"MathJax-Span-1711\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.865em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-1712\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-1713\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"msubsup\" id=\"MathJax-Span-1714\" style=\"padding-left: 0.235em;\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-1715\"><span class=\"mrow\" id=\"MathJax-Span-1716\"><span class=\"mi\" id=\"MathJax-Span-1717\" style=\"font-family: MathJax_Main;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.865em; left: 0.562em;\"><span class=\"mn\" id=\"MathJax-Span-1718\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-1719\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"msubsup\" id=\"MathJax-Span-1720\" style=\"padding-left: 0.235em;\"><span style=\"display: inline-block; position: relative; width: 0.841em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-1721\" style=\"font-family: MathJax_Math-italic;\">c<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.865em; left: 0.422em;\"><span class=\"mn\" id=\"MathJax-Span-1722\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1004.1em, 4.336em, -999.998em); top: -3.959em; left: 50%; margin-left: -2.048em;\"><span class=\"mtd\" id=\"MathJax-Span-1753\"><span class=\"mrow\" id=\"MathJax-Span-1754\"><span class=\"msubsup\" id=\"MathJax-Span-1755\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-1756\"><span class=\"mrow\" id=\"MathJax-Span-1757\"><span class=\"mi\" id=\"MathJax-Span-1758\" style=\"font-family: MathJax_Main;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.865em; left: 0.562em;\"><span class=\"mn\" id=\"MathJax-Span-1759\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"msubsup\" id=\"MathJax-Span-1760\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-1761\"><span class=\"mrow\" id=\"MathJax-Span-1762\"><span class=\"mi\" id=\"MathJax-Span-1763\" style=\"font-family: MathJax_Main;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.865em; left: 0.562em;\"><span class=\"mn\" id=\"MathJax-Span-1764\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-1765\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"msubsup\" id=\"MathJax-Span-1766\" style=\"padding-left: 0.235em;\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-1767\"><span class=\"mrow\" id=\"MathJax-Span-1768\"><span class=\"mi\" id=\"MathJax-Span-1769\" style=\"font-family: MathJax_Main;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.865em; left: 0.468em;\"><span class=\"mn\" id=\"MathJax-Span-1770\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.078em, 1000.89em, 4.476em, -999.998em); top: -2.467em; left: 50%; margin-left: -0.417em;\"><span class=\"mtd\" id=\"MathJax-Span-1793\"><span class=\"mrow\" id=\"MathJax-Span-1794\"><span class=\"msubsup\" id=\"MathJax-Span-1795\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-1796\"><span class=\"mrow\" id=\"MathJax-Span-1797\"><span class=\"mi\" id=\"MathJax-Span-1798\" style=\"font-family: MathJax_Main;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.42em, 4.15em, -999.998em); top: -4.331em; left: 0.468em;\"><span class=\"mn\" id=\"MathJax-Span-1799\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.42em, 4.15em, -999.998em); top: -3.679em; left: 0.468em;\"><span class=\"mn\" id=\"MathJax-Span-1800\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 5.082em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-1801\" style=\"vertical-align: 2.332em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.044em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.467em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.675em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -0.836em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.616em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -2.214em; border-left: 0px solid; width: 0px; height: 4.976em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-47\">\\left|\\begin{array}{ccc}  \\mathrm{a}_1^2+\\mathrm{b}_1+c_1 & \\mathrm{a}_1 \\mathrm{a}_2+\\mathrm{b}_2+c_2 & \\mathrm{a}_1 \\mathrm{a}_3+\\mathrm{b}_3+c_3 \\\\  \\mathrm{b}_1 \\mathrm{b}_2+c_1 & \\mathrm{b}_2^2+\\mathrm{c}_2 & \\mathrm{b}_2 \\mathrm{b}_3+\\mathrm{c}_3 \\\\  \\mathrm{c}_3 c_1 & \\mathrm{c}_3 \\mathrm{c}_2 & \\mathrm{c}_3^2  \\end{array}\\right|<\/script> and \u0394<sub>2<\/sub> = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-48-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-1802\" style=\"width: 6.666em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 5.734em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1005.59em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-1803\"><span class=\"mrow\" id=\"MathJax-Span-1804\"><span class=\"mo\" id=\"MathJax-Span-1805\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-1806\"><span style=\"display: inline-block; position: relative; width: 4.802em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.472em, 1000.93em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.93em, 4.289em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1807\"><span class=\"mrow\" id=\"MathJax-Span-1808\"><span class=\"msubsup\" id=\"MathJax-Span-1809\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-1810\"><span class=\"mrow\" id=\"MathJax-Span-1811\"><span class=\"mi\" id=\"MathJax-Span-1812\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.865em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-1813\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.93em, 4.289em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1828\"><span class=\"mrow\" id=\"MathJax-Span-1829\"><span class=\"msubsup\" id=\"MathJax-Span-1830\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-1831\"><span class=\"mrow\" id=\"MathJax-Span-1832\"><span class=\"mi\" id=\"MathJax-Span-1833\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.865em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-1834\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.93em, 4.336em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1849\"><span class=\"mrow\" id=\"MathJax-Span-1850\"><span class=\"msubsup\" id=\"MathJax-Span-1851\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-1852\"><span class=\"mrow\" id=\"MathJax-Span-1853\"><span class=\"mi\" id=\"MathJax-Span-1854\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.865em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-1855\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.472em, 1000.98em, 6.433em, -999.998em); top: -4.658em; left: 1.913em;\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.98em, 4.289em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1814\"><span class=\"mrow\" id=\"MathJax-Span-1815\"><span class=\"msubsup\" id=\"MathJax-Span-1816\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-1817\"><span class=\"mrow\" id=\"MathJax-Span-1818\"><span class=\"mi\" id=\"MathJax-Span-1819\" style=\"font-family: MathJax_Main;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.865em; left: 0.562em;\"><span class=\"mn\" id=\"MathJax-Span-1820\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.98em, 4.289em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1835\"><span class=\"mrow\" id=\"MathJax-Span-1836\"><span class=\"msubsup\" id=\"MathJax-Span-1837\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-1838\"><span class=\"mrow\" id=\"MathJax-Span-1839\"><span class=\"mi\" id=\"MathJax-Span-1840\" style=\"font-family: MathJax_Main;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.865em; left: 0.562em;\"><span class=\"mn\" id=\"MathJax-Span-1841\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.98em, 4.336em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1856\"><span class=\"mrow\" id=\"MathJax-Span-1857\"><span class=\"msubsup\" id=\"MathJax-Span-1858\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-1859\"><span class=\"mrow\" id=\"MathJax-Span-1860\"><span class=\"mi\" id=\"MathJax-Span-1861\" style=\"font-family: MathJax_Main;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.865em; left: 0.562em;\"><span class=\"mn\" id=\"MathJax-Span-1862\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.662em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.472em, 1000.89em, 6.2em, -999.998em); top: -4.425em; left: 3.917em;\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.89em, 4.289em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1821\"><span class=\"mrow\" id=\"MathJax-Span-1822\"><span class=\"msubsup\" id=\"MathJax-Span-1823\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-1824\"><span class=\"mrow\" id=\"MathJax-Span-1825\"><span class=\"mi\" id=\"MathJax-Span-1826\" style=\"font-family: MathJax_Main;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.865em; left: 0.468em;\"><span class=\"mn\" id=\"MathJax-Span-1827\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.89em, 4.289em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1842\"><span class=\"mrow\" id=\"MathJax-Span-1843\"><span class=\"msubsup\" id=\"MathJax-Span-1844\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-1845\"><span class=\"mrow\" id=\"MathJax-Span-1846\"><span class=\"mi\" id=\"MathJax-Span-1847\" style=\"font-family: MathJax_Main;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.865em; left: 0.468em;\"><span class=\"mn\" id=\"MathJax-Span-1848\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.89em, 4.336em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1863\"><span class=\"mrow\" id=\"MathJax-Span-1864\"><span class=\"msubsup\" id=\"MathJax-Span-1865\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-1866\"><span class=\"mrow\" id=\"MathJax-Span-1867\"><span class=\"mi\" id=\"MathJax-Span-1868\" style=\"font-family: MathJax_Main;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.865em; left: 0.468em;\"><span class=\"mn\" id=\"MathJax-Span-1869\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-1870\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-48\">\\left|\\begin{array}{lll}  \\mathrm{a}_1 & \\mathrm{b}_1 & \\mathrm{c}_1 \\\\  \\mathrm{a}_2 & \\mathrm{b}_2 & \\mathrm{c}_2 \\\\  \\mathrm{a}_3 & \\mathrm{b}_3 & \\mathrm{c}_3  \\end{array}\\right|<\/script>, then find the value of <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-49-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-1871\" style=\"width: 1.54em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.307em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.121em, 1001.31em, 2.938em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-1872\"><span class=\"mfrac\" id=\"MathJax-Span-1873\"><span style=\"display: inline-block; position: relative; width: 1.027em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.89em, 4.243em, -999.998em); top: -4.518em; left: 50%; margin-left: -0.464em;\"><span class=\"msubsup\" id=\"MathJax-Span-1874\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.56em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-1875\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u0394<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.912em; left: 0.608em;\"><span class=\"mn\" id=\"MathJax-Span-1876\" style=\"font-size: 50%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1000.89em, 4.243em, -999.998em); top: -3.586em; left: 50%; margin-left: -0.464em;\"><span class=\"msubsup\" id=\"MathJax-Span-1877\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.56em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-1878\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u0394<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.912em; left: 0.608em;\"><span class=\"mn\" id=\"MathJax-Span-1879\" style=\"font-size: 50%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.03em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.027em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.646em; border-left: 0px solid; width: 0px; height: 1.895em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-49\">\\frac{\\Delta_1}{\\Delta_2}<\/script>.<\/h3>\n\n\n\n<p>Answer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-5963\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3d-8.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(d) 8\" width=\"325\" height=\"1396\" sizes=\"auto, (max-width: 325px) 100vw, 325px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(d) 8\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(d)\"><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Question 7.<br> If \u0394<sub>1<\/sub> = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-50-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-1880\" style=\"width: 10.86em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 9.369em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1009.23em, 6.247em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-1881\"><span class=\"mrow\" id=\"MathJax-Span-1882\"><span class=\"mo\" id=\"MathJax-Span-1883\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.417em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.349em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-1884\"><span style=\"display: inline-block; position: relative; width: 8.437em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.472em, 1002.15em, 6.433em, -999.998em); top: -4.658em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 2.192em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-1885\"><span class=\"mrow\" id=\"MathJax-Span-1886\"><span class=\"mn\" id=\"MathJax-Span-1887\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1002.15em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -1.069em;\"><span class=\"mtd\" id=\"MathJax-Span-1898\"><span class=\"mrow\" id=\"MathJax-Span-1899\"><span class=\"mi\" id=\"MathJax-Span-1900\" style=\"font-family: MathJax_Main;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-1901\"><\/span><span class=\"mi\" id=\"MathJax-Span-1902\" style=\"font-family: MathJax_Math-italic; padding-left: 0.189em;\">\u03b1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1002.1em, 4.336em, -999.998em); top: -2.561em; left: 50%; margin-left: -1.069em;\"><span class=\"mtd\" id=\"MathJax-Span-1911\"><span class=\"mrow\" id=\"MathJax-Span-1912\"><span class=\"mi\" id=\"MathJax-Span-1913\" style=\"font-family: MathJax_Main;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-1914\"><\/span><span class=\"mi\" id=\"MathJax-Span-1915\" style=\"font-family: MathJax_Math-italic; padding-left: 0.189em;\">\u03b2<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.662em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.519em, 1002.15em, 6.293em, -999.998em); top: -4.471em; left: 3.171em;\"><span style=\"display: inline-block; position: relative; width: 2.192em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1002.15em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -1.069em;\"><span class=\"mtd\" id=\"MathJax-Span-1888\"><span class=\"mrow\" id=\"MathJax-Span-1889\"><span class=\"mi\" id=\"MathJax-Span-1890\" style=\"font-family: MathJax_Main;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-1891\"><\/span><span class=\"mi\" id=\"MathJax-Span-1892\" style=\"font-family: MathJax_Math-italic; padding-left: 0.189em;\">\u03b1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-1903\"><span class=\"mrow\" id=\"MathJax-Span-1904\"><span class=\"mn\" id=\"MathJax-Span-1905\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1002.05em, 4.383em, -999.998em); top: -2.561em; left: 50%; margin-left: -1.023em;\"><span class=\"mtd\" id=\"MathJax-Span-1916\"><span class=\"mrow\" id=\"MathJax-Span-1917\"><span class=\"mi\" id=\"MathJax-Span-1918\" style=\"font-family: MathJax_Main;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-1919\"><\/span><span class=\"mi\" id=\"MathJax-Span-1920\" style=\"font-family: MathJax_Math-italic; padding-left: 0.189em;\">\u03b3<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.049em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1002.1em, 6.107em, -999.998em); top: -4.518em; left: 6.34em;\"><span style=\"display: inline-block; position: relative; width: 2.099em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1002.1em, 4.336em, -999.998em); top: -5.356em; left: 50%; margin-left: -1.069em;\"><span class=\"mtd\" id=\"MathJax-Span-1893\"><span class=\"mrow\" id=\"MathJax-Span-1894\"><span class=\"mi\" id=\"MathJax-Span-1895\" style=\"font-family: MathJax_Main;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-1896\"><\/span><span class=\"mi\" id=\"MathJax-Span-1897\" style=\"font-family: MathJax_Math-italic; padding-left: 0.189em;\">\u03b2<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1002.05em, 4.383em, -999.998em); top: -3.959em; left: 50%; margin-left: -1.023em;\"><span class=\"mtd\" id=\"MathJax-Span-1906\"><span class=\"mrow\" id=\"MathJax-Span-1907\"><span class=\"mi\" id=\"MathJax-Span-1908\" style=\"font-family: MathJax_Main;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-1909\"><\/span><span class=\"mi\" id=\"MathJax-Span-1910\" style=\"font-family: MathJax_Math-italic; padding-left: 0.189em;\">\u03b3<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.049em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-1921\"><span class=\"mrow\" id=\"MathJax-Span-1922\"><span class=\"mn\" id=\"MathJax-Span-1923\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.522em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-1924\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.417em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.349em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.997em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-50\">\\left|\\begin{array}{ccc}  1 & \\cos \\alpha & \\cos \\beta \\\\  \\cos \\alpha & 1 & \\cos \\gamma \\\\  \\cos \\beta & \\cos \\gamma & 1  \\end{array}\\right|<\/script> and \u0394<sub>2<\/sub> = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-51-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-1925\" style=\"width: 10.86em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 9.369em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1009.23em, 6.247em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-1926\"><span class=\"mrow\" id=\"MathJax-Span-1927\"><span class=\"mo\" id=\"MathJax-Span-1928\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.417em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.349em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-1929\"><span style=\"display: inline-block; position: relative; width: 8.437em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.472em, 1002.15em, 6.433em, -999.998em); top: -4.658em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 2.192em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-1930\"><span class=\"mrow\" id=\"MathJax-Span-1931\"><span class=\"mn\" id=\"MathJax-Span-1932\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1002.15em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -1.069em;\"><span class=\"mtd\" id=\"MathJax-Span-1943\"><span class=\"mrow\" id=\"MathJax-Span-1944\"><span class=\"mi\" id=\"MathJax-Span-1945\" style=\"font-family: MathJax_Main;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-1946\"><\/span><span class=\"mi\" id=\"MathJax-Span-1947\" style=\"font-family: MathJax_Math-italic; padding-left: 0.189em;\">\u03b1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1002.1em, 4.336em, -999.998em); top: -2.561em; left: 50%; margin-left: -1.069em;\"><span class=\"mtd\" id=\"MathJax-Span-1956\"><span class=\"mrow\" id=\"MathJax-Span-1957\"><span class=\"mi\" id=\"MathJax-Span-1958\" style=\"font-family: MathJax_Main;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-1959\"><\/span><span class=\"mi\" id=\"MathJax-Span-1960\" style=\"font-family: MathJax_Math-italic; padding-left: 0.189em;\">\u03b2<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.662em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.519em, 1002.15em, 6.293em, -999.998em); top: -4.471em; left: 3.171em;\"><span style=\"display: inline-block; position: relative; width: 2.192em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1002.15em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -1.069em;\"><span class=\"mtd\" id=\"MathJax-Span-1933\"><span class=\"mrow\" id=\"MathJax-Span-1934\"><span class=\"mi\" id=\"MathJax-Span-1935\" style=\"font-family: MathJax_Main;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-1936\"><\/span><span class=\"mi\" id=\"MathJax-Span-1937\" style=\"font-family: MathJax_Math-italic; padding-left: 0.189em;\">\u03b1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-1948\"><span class=\"mrow\" id=\"MathJax-Span-1949\"><span class=\"mn\" id=\"MathJax-Span-1950\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1002.05em, 4.383em, -999.998em); top: -2.561em; left: 50%; margin-left: -1.023em;\"><span class=\"mtd\" id=\"MathJax-Span-1961\"><span class=\"mrow\" id=\"MathJax-Span-1962\"><span class=\"mi\" id=\"MathJax-Span-1963\" style=\"font-family: MathJax_Main;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-1964\"><\/span><span class=\"mi\" id=\"MathJax-Span-1965\" style=\"font-family: MathJax_Math-italic; padding-left: 0.189em;\">\u03b3<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.049em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1002.1em, 6.153em, -999.998em); top: -4.518em; left: 6.34em;\"><span style=\"display: inline-block; position: relative; width: 2.099em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1002.1em, 4.336em, -999.998em); top: -5.356em; left: 50%; margin-left: -1.069em;\"><span class=\"mtd\" id=\"MathJax-Span-1938\"><span class=\"mrow\" id=\"MathJax-Span-1939\"><span class=\"mi\" id=\"MathJax-Span-1940\" style=\"font-family: MathJax_Main;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-1941\"><\/span><span class=\"mi\" id=\"MathJax-Span-1942\" style=\"font-family: MathJax_Math-italic; padding-left: 0.189em;\">\u03b2<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1002.05em, 4.383em, -999.998em); top: -3.959em; left: 50%; margin-left: -1.023em;\"><span class=\"mtd\" id=\"MathJax-Span-1951\"><span class=\"mrow\" id=\"MathJax-Span-1952\"><span class=\"mi\" id=\"MathJax-Span-1953\" style=\"font-family: MathJax_Main;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-1954\"><\/span><span class=\"mi\" id=\"MathJax-Span-1955\" style=\"font-family: MathJax_Math-italic; padding-left: 0.189em;\">\u03b3<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.049em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-1966\"><span class=\"mrow\" id=\"MathJax-Span-1967\"><span class=\"mn\" id=\"MathJax-Span-1968\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.522em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-1969\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.417em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.349em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.997em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-51\">\\left|\\begin{array}{ccc}  0 & \\cos \\alpha & \\cos \\beta \\\\  \\cos \\alpha & 0 & \\cos \\gamma \\\\  \\cos \\beta & \\cos \\gamma & 0  \\end{array}\\right|<\/script> and \u0394<sub>1<\/sub> = \u0394<sub>2<\/sub> then show that cos<sup>2<\/sup>\u03b1 + cos<sup>2<\/sup>\u03b2 + cos<sup>2<\/sup>\u03b3 = 1.<\/h3>\n\n\n\n<p>Answer:<br> Given <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-52-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-1970\" style=\"width: 10.86em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 9.369em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1009.23em, 6.247em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-1971\"><span class=\"mrow\" id=\"MathJax-Span-1972\"><span class=\"mo\" id=\"MathJax-Span-1973\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.417em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.349em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-1974\"><span style=\"display: inline-block; position: relative; width: 8.437em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.472em, 1002.15em, 6.433em, -999.998em); top: -4.658em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 2.192em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-1975\"><span class=\"mrow\" id=\"MathJax-Span-1976\"><span class=\"mn\" id=\"MathJax-Span-1977\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1002.15em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -1.069em;\"><span class=\"mtd\" id=\"MathJax-Span-1988\"><span class=\"mrow\" id=\"MathJax-Span-1989\"><span class=\"mi\" id=\"MathJax-Span-1990\" style=\"font-family: MathJax_Main;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-1991\"><\/span><span class=\"mi\" id=\"MathJax-Span-1992\" style=\"font-family: MathJax_Math-italic; padding-left: 0.189em;\">\u03b1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1002.1em, 4.336em, -999.998em); top: -2.561em; left: 50%; margin-left: -1.069em;\"><span class=\"mtd\" id=\"MathJax-Span-2001\"><span class=\"mrow\" id=\"MathJax-Span-2002\"><span class=\"mi\" id=\"MathJax-Span-2003\" style=\"font-family: MathJax_Main;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-2004\"><\/span><span class=\"mi\" id=\"MathJax-Span-2005\" style=\"font-family: MathJax_Math-italic; padding-left: 0.189em;\">\u03b2<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.662em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.519em, 1002.15em, 6.293em, -999.998em); top: -4.471em; left: 3.171em;\"><span style=\"display: inline-block; position: relative; width: 2.192em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1002.15em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -1.069em;\"><span class=\"mtd\" id=\"MathJax-Span-1978\"><span class=\"mrow\" id=\"MathJax-Span-1979\"><span class=\"mi\" id=\"MathJax-Span-1980\" style=\"font-family: MathJax_Main;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-1981\"><\/span><span class=\"mi\" id=\"MathJax-Span-1982\" style=\"font-family: MathJax_Math-italic; padding-left: 0.189em;\">\u03b1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-1993\"><span class=\"mrow\" id=\"MathJax-Span-1994\"><span class=\"mn\" id=\"MathJax-Span-1995\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1002.05em, 4.383em, -999.998em); top: -2.561em; left: 50%; margin-left: -1.023em;\"><span class=\"mtd\" id=\"MathJax-Span-2006\"><span class=\"mrow\" id=\"MathJax-Span-2007\"><span class=\"mi\" id=\"MathJax-Span-2008\" style=\"font-family: MathJax_Main;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-2009\"><\/span><span class=\"mi\" id=\"MathJax-Span-2010\" style=\"font-family: MathJax_Math-italic; padding-left: 0.189em;\">\u03b3<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.049em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1002.1em, 6.107em, -999.998em); top: -4.518em; left: 6.34em;\"><span style=\"display: inline-block; position: relative; width: 2.099em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1002.1em, 4.336em, -999.998em); top: -5.356em; left: 50%; margin-left: -1.069em;\"><span class=\"mtd\" id=\"MathJax-Span-1983\"><span class=\"mrow\" id=\"MathJax-Span-1984\"><span class=\"mi\" id=\"MathJax-Span-1985\" style=\"font-family: MathJax_Main;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-1986\"><\/span><span class=\"mi\" id=\"MathJax-Span-1987\" style=\"font-family: MathJax_Math-italic; padding-left: 0.189em;\">\u03b2<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1002.05em, 4.383em, -999.998em); top: -3.959em; left: 50%; margin-left: -1.023em;\"><span class=\"mtd\" id=\"MathJax-Span-1996\"><span class=\"mrow\" id=\"MathJax-Span-1997\"><span class=\"mi\" id=\"MathJax-Span-1998\" style=\"font-family: MathJax_Main;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-1999\"><\/span><span class=\"mi\" id=\"MathJax-Span-2000\" style=\"font-family: MathJax_Math-italic; padding-left: 0.189em;\">\u03b3<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.049em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-2011\"><span class=\"mrow\" id=\"MathJax-Span-2012\"><span class=\"mn\" id=\"MathJax-Span-2013\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.522em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-2014\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.417em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.349em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.997em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-52\">\\left|\\begin{array}{ccc}  1 & \\cos \\alpha & \\cos \\beta \\\\  \\cos \\alpha & 1 & \\cos \\gamma \\\\  \\cos \\beta & \\cos \\gamma & 1  \\end{array}\\right|<\/script><br> = (1 \u2013 cos<sup>2<\/sup>\u03b3) \u2013 cos \u03b1 (cos \u03b1 \u2013 cos \u03b2 cos \u03b3) + cos \u03b2 (cos \u03b1 cos \u03b3 \u2013 cos \u03b2)<br> = 1 \u2013 cos<sup>2<\/sup>\u03b3 \u2013 cos<sup>2<\/sup>\u03b1 + cos \u03b2 cos \u03b1 cos \u03b3 + cos \u03b1 cos \u03b2 cos \u03b3 \u2013 cos<sup>2<\/sup>\u03b2<br> = 1 \u2013 (cos<sup>2<\/sup>\u03b1 + cos<sup>2<\/sup>\u03b2 + cos<sup>2<\/sup>\u03b3) + 2 cos \u03b1 cos \u03b2 cos \u03b3<\/p>\n\n\n\n<p>\u0394<sub>2<\/sub> = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-53-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-2015\" style=\"width: 10.86em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 9.369em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1009.23em, 6.247em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-2016\"><span class=\"mrow\" id=\"MathJax-Span-2017\"><span class=\"mo\" id=\"MathJax-Span-2018\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.417em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.349em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-2019\"><span style=\"display: inline-block; position: relative; width: 8.437em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.472em, 1002.15em, 6.433em, -999.998em); top: -4.658em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 2.192em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-2020\"><span class=\"mrow\" id=\"MathJax-Span-2021\"><span class=\"mn\" id=\"MathJax-Span-2022\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1002.15em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -1.069em;\"><span class=\"mtd\" id=\"MathJax-Span-2033\"><span class=\"mrow\" id=\"MathJax-Span-2034\"><span class=\"mi\" id=\"MathJax-Span-2035\" style=\"font-family: MathJax_Main;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-2036\"><\/span><span class=\"mi\" id=\"MathJax-Span-2037\" style=\"font-family: MathJax_Math-italic; padding-left: 0.189em;\">\u03b1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1002.1em, 4.336em, -999.998em); top: -2.561em; left: 50%; margin-left: -1.069em;\"><span class=\"mtd\" id=\"MathJax-Span-2046\"><span class=\"mrow\" id=\"MathJax-Span-2047\"><span class=\"mi\" id=\"MathJax-Span-2048\" style=\"font-family: MathJax_Main;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-2049\"><\/span><span class=\"mi\" id=\"MathJax-Span-2050\" style=\"font-family: MathJax_Math-italic; padding-left: 0.189em;\">\u03b2<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.662em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.519em, 1002.15em, 6.293em, -999.998em); top: -4.471em; left: 3.171em;\"><span style=\"display: inline-block; position: relative; width: 2.192em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1002.15em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -1.069em;\"><span class=\"mtd\" id=\"MathJax-Span-2023\"><span class=\"mrow\" id=\"MathJax-Span-2024\"><span class=\"mi\" id=\"MathJax-Span-2025\" style=\"font-family: MathJax_Main;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-2026\"><\/span><span class=\"mi\" id=\"MathJax-Span-2027\" style=\"font-family: MathJax_Math-italic; padding-left: 0.189em;\">\u03b1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-2038\"><span class=\"mrow\" id=\"MathJax-Span-2039\"><span class=\"mn\" id=\"MathJax-Span-2040\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1002.05em, 4.383em, -999.998em); top: -2.561em; left: 50%; margin-left: -1.023em;\"><span class=\"mtd\" id=\"MathJax-Span-2051\"><span class=\"mrow\" id=\"MathJax-Span-2052\"><span class=\"mi\" id=\"MathJax-Span-2053\" style=\"font-family: MathJax_Main;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-2054\"><\/span><span class=\"mi\" id=\"MathJax-Span-2055\" style=\"font-family: MathJax_Math-italic; padding-left: 0.189em;\">\u03b3<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.049em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1002.1em, 6.153em, -999.998em); top: -4.518em; left: 6.34em;\"><span style=\"display: inline-block; position: relative; width: 2.099em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1002.1em, 4.336em, -999.998em); top: -5.356em; left: 50%; margin-left: -1.069em;\"><span class=\"mtd\" id=\"MathJax-Span-2028\"><span class=\"mrow\" id=\"MathJax-Span-2029\"><span class=\"mi\" id=\"MathJax-Span-2030\" style=\"font-family: MathJax_Main;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-2031\"><\/span><span class=\"mi\" id=\"MathJax-Span-2032\" style=\"font-family: MathJax_Math-italic; padding-left: 0.189em;\">\u03b2<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1002.05em, 4.383em, -999.998em); top: -3.959em; left: 50%; margin-left: -1.023em;\"><span class=\"mtd\" id=\"MathJax-Span-2041\"><span class=\"mrow\" id=\"MathJax-Span-2042\"><span class=\"mi\" id=\"MathJax-Span-2043\" style=\"font-family: MathJax_Main;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-2044\"><\/span><span class=\"mi\" id=\"MathJax-Span-2045\" style=\"font-family: MathJax_Math-italic; padding-left: 0.189em;\">\u03b3<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.049em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-2056\"><span class=\"mrow\" id=\"MathJax-Span-2057\"><span class=\"mn\" id=\"MathJax-Span-2058\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.522em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-2059\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.417em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.349em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.997em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-53\">\\left|\\begin{array}{ccc}  0 & \\cos \\alpha & \\cos \\beta \\\\  \\cos \\alpha & 0 & \\cos \\gamma \\\\  \\cos \\beta & \\cos \\gamma & 0  \\end{array}\\right|<\/script><br> = \u2013 cos \u03b1 (0 \u2013 cos \u03b3 cos \u03b2) + cos \u03b2 (cos \u03b1 cos \u03b3)<br> = cos \u03b1 cos \u03b2 cos \u03b3 + cos \u03b1 cos \u03b2 cos \u03b3<br> = 2cos \u03b1 cos \u03b2 cos \u03b3<br> Also given \u0394<sub>1<\/sub> = \u0394<sub>2<\/sub><br> \u21d2 1 \u2013 (cos<sup>2<\/sup>\u03b1 + cos<sup>2<\/sup>\u03b2 + cos<sup>2<\/sup>\u03b3) + 2 cos \u03b1 cos \u03b2 cos \u03b3<br> = 2 cos \u03b1 cos \u03b2 cos \u03b3<br> \u21d2 1 \u2013 (cos<sup>2<\/sup>\u03b1 + cos<sup>2<\/sup>\u03b2 + cos<sup>2<\/sup>\u03b3) = 0<br> \u2234 cos<sup>2<\/sup>\u03b1 + cos<sup>2<\/sup>\u03b2 + cos<sup>2<\/sup>\u03b3 = 1<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">III.<br> Question 1.<br> Show that<br> <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-54-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-2060\" style=\"width: 18.642em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 16.079em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1015.94em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-2061\"><span class=\"mrow\" id=\"MathJax-Span-2062\"><span class=\"mo\" id=\"MathJax-Span-2063\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-2064\"><span style=\"display: inline-block; position: relative; width: 15.147em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1004.38em, 6.107em, -999.998em); top: -4.518em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 4.383em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1004.38em, 4.243em, -999.998em); top: -5.356em; left: 50%; margin-left: -2.188em;\"><span class=\"mtd\" id=\"MathJax-Span-2065\"><span class=\"mrow\" id=\"MathJax-Span-2066\"><span class=\"mi\" id=\"MathJax-Span-2067\" style=\"font-family: MathJax_Math-italic;\">a<\/span><span class=\"mo\" id=\"MathJax-Span-2068\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mi\" id=\"MathJax-Span-2069\" style=\"font-family: MathJax_Math-italic; padding-left: 0.235em;\">b<\/span><span class=\"mo\" id=\"MathJax-Span-2070\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mn\" id=\"MathJax-Span-2071\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">2<\/span><span class=\"mi\" id=\"MathJax-Span-2072\" style=\"font-family: MathJax_Math-italic;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.42em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-2079\"><span class=\"mrow\" id=\"MathJax-Span-2080\"><span class=\"mi\" id=\"MathJax-Span-2081\" style=\"font-family: MathJax_Math-italic;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.42em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-2093\"><span class=\"mrow\" id=\"MathJax-Span-2094\"><span class=\"mi\" id=\"MathJax-Span-2095\" style=\"font-family: MathJax_Math-italic;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.522em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1004.38em, 5.827em, -999.998em); top: -4.238em; left: 5.408em;\"><span style=\"display: inline-block; position: relative; width: 4.383em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.277em;\"><span class=\"mtd\" id=\"MathJax-Span-2073\"><span class=\"mrow\" id=\"MathJax-Span-2074\"><span class=\"mi\" id=\"MathJax-Span-2075\" style=\"font-family: MathJax_Math-italic;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1004.38em, 4.243em, -999.998em); top: -3.959em; left: 50%; margin-left: -2.188em;\"><span class=\"mtd\" id=\"MathJax-Span-2082\"><span class=\"mrow\" id=\"MathJax-Span-2083\"><span class=\"mi\" id=\"MathJax-Span-2084\" style=\"font-family: MathJax_Math-italic;\">b<\/span><span class=\"mo\" id=\"MathJax-Span-2085\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mi\" id=\"MathJax-Span-2086\" style=\"font-family: MathJax_Math-italic; padding-left: 0.235em;\">c<\/span><span class=\"mo\" id=\"MathJax-Span-2087\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mn\" id=\"MathJax-Span-2088\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">2<\/span><span class=\"mi\" id=\"MathJax-Span-2089\" style=\"font-family: MathJax_Math-italic;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.277em;\"><span class=\"mtd\" id=\"MathJax-Span-2096\"><span class=\"mrow\" id=\"MathJax-Span-2097\"><span class=\"mi\" id=\"MathJax-Span-2098\" style=\"font-family: MathJax_Math-italic;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.243em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.379em, 1004.38em, 6.247em, -999.998em); top: -4.564em; left: 10.767em;\"><span style=\"display: inline-block; position: relative; width: 4.383em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.42em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-2076\"><span class=\"mrow\" id=\"MathJax-Span-2077\"><span class=\"mi\" id=\"MathJax-Span-2078\" style=\"font-family: MathJax_Math-italic;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.42em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-2090\"><span class=\"mrow\" id=\"MathJax-Span-2091\"><span class=\"mi\" id=\"MathJax-Span-2092\" style=\"font-family: MathJax_Math-italic;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1004.38em, 4.243em, -999.998em); top: -2.561em; left: 50%; margin-left: -2.188em;\"><span class=\"mtd\" id=\"MathJax-Span-2099\"><span class=\"mrow\" id=\"MathJax-Span-2100\"><span class=\"mi\" id=\"MathJax-Span-2101\" style=\"font-family: MathJax_Math-italic;\">c<\/span><span class=\"mo\" id=\"MathJax-Span-2102\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mi\" id=\"MathJax-Span-2103\" style=\"font-family: MathJax_Math-italic; padding-left: 0.235em;\">a<\/span><span class=\"mo\" id=\"MathJax-Span-2104\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mn\" id=\"MathJax-Span-2105\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">2<\/span><span class=\"mi\" id=\"MathJax-Span-2106\" style=\"font-family: MathJax_Math-italic;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-2107\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-54\">\\left|\\begin{array}{ccc}  a+b+2 c & a & b \\\\  c & b+c+2 a & b \\\\  c & a & c+a+2 b  \\end{array}\\right|<\/script> = 2(a + b + c)<sup>3<\/sup><\/h3>\n\n\n\n<p>Answer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-5962\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3d-9.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(d) 9\" width=\"300\" height=\"446\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(d) 9\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(d)\"><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Question 2.<br> Show that <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-55-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-2108\" style=\"width: 5.687em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.895em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(-0.184em, 1004.89em, 4.056em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-2109\"><span class=\"msubsup\" id=\"MathJax-Span-2110\"><span style=\"display: inline-block; position: relative; width: 4.895em; height: 0px;\"><span style=\"position: absolute; clip: rect(2.519em, 1004.38em, 6.573em, -999.998em); top: -4.797em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-2111\"><span class=\"mo\" id=\"MathJax-Span-2112\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-2113\"><span style=\"display: inline-block; position: relative; width: 3.59em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.51em, 5.827em, -999.998em); top: -4.238em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-2114\"><span class=\"mrow\" id=\"MathJax-Span-2115\"><span class=\"mi\" id=\"MathJax-Span-2116\" style=\"font-family: MathJax_Math-italic;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.42em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-2123\"><span class=\"mrow\" id=\"MathJax-Span-2124\"><span class=\"mi\" id=\"MathJax-Span-2125\" style=\"font-family: MathJax_Math-italic;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.42em, 4.15em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-2132\"><span class=\"mrow\" id=\"MathJax-Span-2133\"><span class=\"mi\" id=\"MathJax-Span-2134\" style=\"font-family: MathJax_Math-italic;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.243em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.51em, 6.107em, -999.998em); top: -4.518em; left: 1.54em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.42em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-2117\"><span class=\"mrow\" id=\"MathJax-Span-2118\"><span class=\"mi\" id=\"MathJax-Span-2119\" style=\"font-family: MathJax_Math-italic;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.42em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-2126\"><span class=\"mrow\" id=\"MathJax-Span-2127\"><span class=\"mi\" id=\"MathJax-Span-2128\" style=\"font-family: MathJax_Math-italic;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.15em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-2135\"><span class=\"mrow\" id=\"MathJax-Span-2136\"><span class=\"mi\" id=\"MathJax-Span-2137\" style=\"font-family: MathJax_Math-italic;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.522em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.51em, 5.827em, -999.998em); top: -4.238em; left: 3.078em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.42em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-2120\"><span class=\"mrow\" id=\"MathJax-Span-2121\"><span class=\"mi\" id=\"MathJax-Span-2122\" style=\"font-family: MathJax_Math-italic;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-2129\"><span class=\"mrow\" id=\"MathJax-Span-2130\"><span class=\"mi\" id=\"MathJax-Span-2131\" style=\"font-family: MathJax_Math-italic;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.42em, 4.15em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-2138\"><span class=\"mrow\" id=\"MathJax-Span-2139\"><span class=\"mi\" id=\"MathJax-Span-2140\" style=\"font-family: MathJax_Math-italic;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.243em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-2141\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.802em;\"><\/span><\/span><span style=\"position: absolute; top: -5.869em; left: 4.476em;\"><span class=\"mn\" id=\"MathJax-Span-2142\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.759em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-55\">\\left|\\begin{array}{lll}  a & b & c \\\\  b & c & a \\\\  c & a & b  \\end{array}\\right|^2<\/script> = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-56-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-2143\" style=\"width: 15.846em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 13.656em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1013.52em, 6.293em, -999.998em); top: -4.471em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-2144\"><span class=\"mrow\" id=\"MathJax-Span-2145\"><span class=\"mo\" id=\"MathJax-Span-2146\" style=\"vertical-align: 2.192em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.37em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.514em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.815em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.069em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-2147\"><span style=\"display: inline-block; position: relative; width: 12.724em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.379em, 1003.59em, 6.34em, -999.998em); top: -4.704em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 3.59em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.031em, 1003.59em, 4.243em, -999.998em); top: -5.356em; left: 50%; margin-left: -1.768em;\"><span class=\"mtd\" id=\"MathJax-Span-2148\"><span class=\"mrow\" id=\"MathJax-Span-2149\"><span class=\"mn\" id=\"MathJax-Span-2150\" style=\"font-family: MathJax_Main;\">2<\/span><span class=\"mi\" id=\"MathJax-Span-2151\" style=\"font-family: MathJax_Math-italic;\">b<\/span><span class=\"mi\" id=\"MathJax-Span-2152\" style=\"font-family: MathJax_Math-italic;\">c<\/span><span class=\"mo\" id=\"MathJax-Span-2153\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"msubsup\" id=\"MathJax-Span-2154\" style=\"padding-left: 0.235em;\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-2155\" style=\"font-family: MathJax_Math-italic;\">a<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-2156\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.031em, 1000.84em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.417em;\"><span class=\"mtd\" id=\"MathJax-Span-2167\"><span class=\"mrow\" id=\"MathJax-Span-2168\"><span class=\"msubsup\" id=\"MathJax-Span-2169\"><span style=\"display: inline-block; position: relative; width: 0.841em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-2170\" style=\"font-family: MathJax_Math-italic;\">c<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.422em;\"><span class=\"mn\" id=\"MathJax-Span-2171\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.031em, 1000.84em, 4.15em, -999.998em); top: -2.514em; left: 50%; margin-left: -0.417em;\"><span class=\"mtd\" id=\"MathJax-Span-2186\"><span class=\"mrow\" id=\"MathJax-Span-2187\"><span class=\"msubsup\" id=\"MathJax-Span-2188\"><span style=\"display: inline-block; position: relative; width: 0.841em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-2189\" style=\"font-family: MathJax_Math-italic;\">b<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.422em;\"><span class=\"mn\" id=\"MathJax-Span-2190\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.709em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.379em, 1003.59em, 6.34em, -999.998em); top: -4.704em; left: 4.569em;\"><span style=\"display: inline-block; position: relative; width: 3.59em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.031em, 1000.84em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.417em;\"><span class=\"mtd\" id=\"MathJax-Span-2157\"><span class=\"mrow\" id=\"MathJax-Span-2158\"><span class=\"msubsup\" id=\"MathJax-Span-2159\"><span style=\"display: inline-block; position: relative; width: 0.841em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-2160\" style=\"font-family: MathJax_Math-italic;\">c<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.422em;\"><span class=\"mn\" id=\"MathJax-Span-2161\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.031em, 1003.59em, 4.243em, -999.998em); top: -3.959em; left: 50%; margin-left: -1.768em;\"><span class=\"mtd\" id=\"MathJax-Span-2172\"><span class=\"mrow\" id=\"MathJax-Span-2173\"><span class=\"mn\" id=\"MathJax-Span-2174\" style=\"font-family: MathJax_Main;\">2<\/span><span class=\"mi\" id=\"MathJax-Span-2175\" style=\"font-family: MathJax_Math-italic;\">a<\/span><span class=\"mi\" id=\"MathJax-Span-2176\" style=\"font-family: MathJax_Math-italic;\">c<\/span><span class=\"mo\" id=\"MathJax-Span-2177\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"msubsup\" id=\"MathJax-Span-2178\" style=\"padding-left: 0.235em;\"><span style=\"display: inline-block; position: relative; width: 0.841em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-2179\" style=\"font-family: MathJax_Math-italic;\">b<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.422em;\"><span class=\"mn\" id=\"MathJax-Span-2180\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.031em, 1000.98em, 4.15em, -999.998em); top: -2.514em; left: 50%; margin-left: -0.464em;\"><span class=\"mtd\" id=\"MathJax-Span-2191\"><span class=\"mrow\" id=\"MathJax-Span-2192\"><span class=\"msubsup\" id=\"MathJax-Span-2193\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-2194\" style=\"font-family: MathJax_Math-italic;\">a<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-2195\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.709em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.472em, 1003.59em, 6.526em, -999.998em); top: -4.797em; left: 9.136em;\"><span style=\"display: inline-block; position: relative; width: 3.59em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.031em, 1000.84em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.417em;\"><span class=\"mtd\" id=\"MathJax-Span-2162\"><span class=\"mrow\" id=\"MathJax-Span-2163\"><span class=\"msubsup\" id=\"MathJax-Span-2164\"><span style=\"display: inline-block; position: relative; width: 0.841em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-2165\" style=\"font-family: MathJax_Math-italic;\">b<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.422em;\"><span class=\"mn\" id=\"MathJax-Span-2166\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.031em, 1000.98em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.464em;\"><span class=\"mtd\" id=\"MathJax-Span-2181\"><span class=\"mrow\" id=\"MathJax-Span-2182\"><span class=\"msubsup\" id=\"MathJax-Span-2183\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-2184\" style=\"font-family: MathJax_Math-italic;\">a<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-2185\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.031em, 1003.59em, 4.243em, -999.998em); top: -2.514em; left: 50%; margin-left: -1.768em;\"><span class=\"mtd\" id=\"MathJax-Span-2196\"><span class=\"mrow\" id=\"MathJax-Span-2197\"><span class=\"mn\" id=\"MathJax-Span-2198\" style=\"font-family: MathJax_Main;\">2<\/span><span class=\"mi\" id=\"MathJax-Span-2199\" style=\"font-family: MathJax_Math-italic;\">a<\/span><span class=\"mi\" id=\"MathJax-Span-2200\" style=\"font-family: MathJax_Math-italic;\">b<\/span><span class=\"mo\" id=\"MathJax-Span-2201\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"msubsup\" id=\"MathJax-Span-2202\" style=\"padding-left: 0.235em;\"><span style=\"display: inline-block; position: relative; width: 0.841em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-2203\" style=\"font-family: MathJax_Math-italic;\">c<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.422em;\"><span class=\"mn\" id=\"MathJax-Span-2204\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.802em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-2205\" style=\"vertical-align: 2.192em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.37em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.514em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.815em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.069em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.997em; border-left: 0px solid; width: 0px; height: 4.651em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-56\">\\left|\\begin{array}{ccc}  2 b c-a^2 & c^2 & b^2 \\\\  c^2 & 2 a c-b^2 & a^2 \\\\  b^2 & a^2 & 2 a b-c^2  \\end{array}\\right|<\/script> = (a<sup>3<\/sup> + b<sup>3<\/sup> + c<sup>3<\/sup> \u2013 3abc)<sup>2<\/sup>. (May 2014, Mar. 01\u2032)<\/h3>\n\n\n\n<p>Answer:<br> Let \u0394 = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-57-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-2206\" style=\"width: 5.268em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.522em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1004.38em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-2207\"><span class=\"mrow\" id=\"MathJax-Span-2208\"><span class=\"mo\" id=\"MathJax-Span-2209\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-2210\"><span style=\"display: inline-block; position: relative; width: 3.59em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.51em, 5.827em, -999.998em); top: -4.238em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-2211\"><span class=\"mrow\" id=\"MathJax-Span-2212\"><span class=\"mi\" id=\"MathJax-Span-2213\" style=\"font-family: MathJax_Math-italic;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.42em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-2220\"><span class=\"mrow\" id=\"MathJax-Span-2221\"><span class=\"mi\" id=\"MathJax-Span-2222\" style=\"font-family: MathJax_Math-italic;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.42em, 4.15em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-2229\"><span class=\"mrow\" id=\"MathJax-Span-2230\"><span class=\"mi\" id=\"MathJax-Span-2231\" style=\"font-family: MathJax_Math-italic;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.243em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.51em, 6.107em, -999.998em); top: -4.518em; left: 1.54em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.42em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-2214\"><span class=\"mrow\" id=\"MathJax-Span-2215\"><span class=\"mi\" id=\"MathJax-Span-2216\" style=\"font-family: MathJax_Math-italic;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.42em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-2223\"><span class=\"mrow\" id=\"MathJax-Span-2224\"><span class=\"mi\" id=\"MathJax-Span-2225\" style=\"font-family: MathJax_Math-italic;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.15em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-2232\"><span class=\"mrow\" id=\"MathJax-Span-2233\"><span class=\"mi\" id=\"MathJax-Span-2234\" style=\"font-family: MathJax_Math-italic;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.522em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.51em, 5.827em, -999.998em); top: -4.238em; left: 3.078em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.42em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-2217\"><span class=\"mrow\" id=\"MathJax-Span-2218\"><span class=\"mi\" id=\"MathJax-Span-2219\" style=\"font-family: MathJax_Math-italic;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-2226\"><span class=\"mrow\" id=\"MathJax-Span-2227\"><span class=\"mi\" id=\"MathJax-Span-2228\" style=\"font-family: MathJax_Math-italic;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.42em, 4.15em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-2235\"><span class=\"mrow\" id=\"MathJax-Span-2236\"><span class=\"mi\" id=\"MathJax-Span-2237\" style=\"font-family: MathJax_Math-italic;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.243em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-2238\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-57\">\\left|\\begin{array}{lll}  a & b & c \\\\  b & c & a \\\\  c & a & b  \\end{array}\\right|<\/script> = a(bc \u2013 a<sup>2<\/sup>) \u2013 b(b<sup>2<\/sup> \u2013 ac) + c(ab \u2013 c<sup>2<\/sup>)<br> = abc \u2013 a<sup>3<\/sup> \u2013 b<sup>3<\/sup> + abc + abc \u2013 c<sup>3<\/sup><br> = \u2013 (a<sup>3<\/sup> + b<sup>3<\/sup> + c<sup>3<\/sup> \u2013 3abc)<br> \u21d2 \u0394<sup>2<\/sup> = (a<sup>3<\/sup> + b<sup>3<\/sup> + c<sup>3<\/sup> \u2013 3abc)<sup>2<\/sup> \u2026\u2026\u2026\u2026..(1)<br>\n\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-5961\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3d-10.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(d) 10\" width=\"349\" height=\"536\" sizes=\"auto, (max-width: 349px) 100vw, 349px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(d) 10\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(d)\">\n\n<br> From (1) and (2) the result is proved.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Question 3.<br> Show that <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-58-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-2239\" style=\"width: 11em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 9.462em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1009.32em, 6.293em, -999.998em); top: -4.471em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-2240\"><span class=\"mrow\" id=\"MathJax-Span-2241\"><span class=\"mo\" id=\"MathJax-Span-2242\" style=\"vertical-align: 2.192em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.417em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.349em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-2243\"><span style=\"display: inline-block; position: relative; width: 8.53em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1003.22em, 6.293em, -999.998em); top: -4.658em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 3.218em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.031em, 1003.22em, 4.243em, -999.998em); top: -5.356em; left: 50%; margin-left: -1.629em;\"><span class=\"mtd\" id=\"MathJax-Span-2244\"><span class=\"mrow\" id=\"MathJax-Span-2245\"><span class=\"msubsup\" id=\"MathJax-Span-2246\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-2247\" style=\"font-family: MathJax_Math-italic;\">a<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-2248\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-2249\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mn\" id=\"MathJax-Span-2250\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">2<\/span><span class=\"mi\" id=\"MathJax-Span-2251\" style=\"font-family: MathJax_Math-italic;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1002.71em, 4.243em, -999.998em); top: -3.959em; left: 50%; margin-left: -1.396em;\"><span class=\"mtd\" id=\"MathJax-Span-2261\"><span class=\"mrow\" id=\"MathJax-Span-2262\"><span class=\"mn\" id=\"MathJax-Span-2263\" style=\"font-family: MathJax_Main;\">2<\/span><span class=\"mi\" id=\"MathJax-Span-2264\" style=\"font-family: MathJax_Math-italic;\">a<\/span><span class=\"mo\" id=\"MathJax-Span-2265\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mn\" id=\"MathJax-Span-2266\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-2275\"><span class=\"mrow\" id=\"MathJax-Span-2276\"><span class=\"mn\" id=\"MathJax-Span-2277\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.662em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1002.71em, 6.107em, -999.998em); top: -4.471em; left: 4.243em;\"><span style=\"display: inline-block; position: relative; width: 2.798em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1002.71em, 4.243em, -999.998em); top: -5.356em; left: 50%; margin-left: -1.396em;\"><span class=\"mtd\" id=\"MathJax-Span-2252\"><span class=\"mrow\" id=\"MathJax-Span-2253\"><span class=\"mn\" id=\"MathJax-Span-2254\" style=\"font-family: MathJax_Main;\">2<\/span><span class=\"mi\" id=\"MathJax-Span-2255\" style=\"font-family: MathJax_Math-italic;\">a<\/span><span class=\"mo\" id=\"MathJax-Span-2256\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mn\" id=\"MathJax-Span-2257\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1002.24em, 4.243em, -999.998em); top: -3.959em; left: 50%; margin-left: -1.116em;\"><span class=\"mtd\" id=\"MathJax-Span-2267\"><span class=\"mrow\" id=\"MathJax-Span-2268\"><span class=\"mi\" id=\"MathJax-Span-2269\" style=\"font-family: MathJax_Math-italic;\">a<\/span><span class=\"mo\" id=\"MathJax-Span-2270\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mn\" id=\"MathJax-Span-2271\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-2278\"><span class=\"mrow\" id=\"MathJax-Span-2279\"><span class=\"mn\" id=\"MathJax-Span-2280\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.42em, 6.06em, -999.998em); top: -4.471em; left: 8.017em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-2258\"><span class=\"mrow\" id=\"MathJax-Span-2259\"><span class=\"mn\" id=\"MathJax-Span-2260\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-2272\"><span class=\"mrow\" id=\"MathJax-Span-2273\"><span class=\"mn\" id=\"MathJax-Span-2274\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-2281\"><span class=\"mrow\" id=\"MathJax-Span-2282\"><span class=\"mn\" id=\"MathJax-Span-2283\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-2284\" style=\"vertical-align: 2.192em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.417em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.349em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.997em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-58\">\\left|\\begin{array}{ccc}  a^2+2 a & 2 a+1 & 1 \\\\  2 a+1 & a+2 & 1 \\\\  3 & 3 & 1  \\end{array}\\right|<\/script> = (a \u2013 1)<sup>3<\/sup>. (March 2007)<\/h3>\n\n\n\n<p>Answer:<br> Apply operations R<sub>1<\/sub> \u2013 R<sub>2<\/sub> and R<sub>2<\/sub> \u2013 R<sub>3<\/sub> we get<br>\n\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-5960\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3d-11.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(d) 11\" width=\"233\" height=\"249\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(d) 11\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(d)\"><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Question 4.<br> Show that <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-59-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-2285\" style=\"width: 6.666em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 5.734em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1005.59em, 6.386em, -999.998em); top: -4.518em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-2286\"><span class=\"mrow\" id=\"MathJax-Span-2287\"><span class=\"mo\" id=\"MathJax-Span-2288\" style=\"vertical-align: 2.239em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.277em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.514em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.768em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.023em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-2289\"><span style=\"display: inline-block; position: relative; width: 4.802em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.379em, 1000.93em, 6.107em, -999.998em); top: -4.425em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.51em, 4.15em, -999.998em); top: -5.45em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-2290\"><span class=\"mrow\" id=\"MathJax-Span-2291\"><span class=\"texatom\" id=\"MathJax-Span-2292\"><span class=\"mrow\" id=\"MathJax-Span-2293\"><span class=\"mi\" id=\"MathJax-Span-2294\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.031em, 1000.93em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.464em;\"><span class=\"mtd\" id=\"MathJax-Span-2305\"><span class=\"mrow\" id=\"MathJax-Span-2306\"><span class=\"msubsup\" id=\"MathJax-Span-2307\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-2308\"><span class=\"mrow\" id=\"MathJax-Span-2309\"><span class=\"mi\" id=\"MathJax-Span-2310\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-2311\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.031em, 1000.93em, 4.15em, -999.998em); top: -2.467em; left: 50%; margin-left: -0.464em;\"><span class=\"mtd\" id=\"MathJax-Span-2326\"><span class=\"mrow\" id=\"MathJax-Span-2327\"><span class=\"msubsup\" id=\"MathJax-Span-2328\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-2329\"><span class=\"mrow\" id=\"MathJax-Span-2330\"><span class=\"mi\" id=\"MathJax-Span-2331\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-2332\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.425em, 1000.98em, 6.386em, -999.998em); top: -4.704em; left: 1.913em;\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.51em, 4.15em, -999.998em); top: -5.45em; left: 50%; margin-left: -0.277em;\"><span class=\"mtd\" id=\"MathJax-Span-2295\"><span class=\"mrow\" id=\"MathJax-Span-2296\"><span class=\"texatom\" id=\"MathJax-Span-2297\"><span class=\"mrow\" id=\"MathJax-Span-2298\"><span class=\"mi\" id=\"MathJax-Span-2299\" style=\"font-family: MathJax_Main;\">b<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.985em, 1000.98em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.51em;\"><span class=\"mtd\" id=\"MathJax-Span-2312\"><span class=\"mrow\" id=\"MathJax-Span-2313\"><span class=\"msubsup\" id=\"MathJax-Span-2314\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-2315\"><span class=\"mrow\" id=\"MathJax-Span-2316\"><span class=\"mi\" id=\"MathJax-Span-2317\" style=\"font-family: MathJax_Main;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.425em; left: 0.562em;\"><span class=\"mn\" id=\"MathJax-Span-2318\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.985em, 1000.98em, 4.15em, -999.998em); top: -2.467em; left: 50%; margin-left: -0.51em;\"><span class=\"mtd\" id=\"MathJax-Span-2333\"><span class=\"mrow\" id=\"MathJax-Span-2334\"><span class=\"msubsup\" id=\"MathJax-Span-2335\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-2336\"><span class=\"mrow\" id=\"MathJax-Span-2337\"><span class=\"mi\" id=\"MathJax-Span-2338\" style=\"font-family: MathJax_Main;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.425em; left: 0.562em;\"><span class=\"mn\" id=\"MathJax-Span-2339\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.709em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.379em, 1000.89em, 6.107em, -999.998em); top: -4.425em; left: 3.917em;\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.42em, 4.15em, -999.998em); top: -5.45em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-2300\"><span class=\"mrow\" id=\"MathJax-Span-2301\"><span class=\"texatom\" id=\"MathJax-Span-2302\"><span class=\"mrow\" id=\"MathJax-Span-2303\"><span class=\"mi\" id=\"MathJax-Span-2304\" style=\"font-family: MathJax_Main;\">c<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.031em, 1000.89em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.417em;\"><span class=\"mtd\" id=\"MathJax-Span-2319\"><span class=\"mrow\" id=\"MathJax-Span-2320\"><span class=\"msubsup\" id=\"MathJax-Span-2321\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-2322\"><span class=\"mrow\" id=\"MathJax-Span-2323\"><span class=\"mi\" id=\"MathJax-Span-2324\" style=\"font-family: MathJax_Main;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.468em;\"><span class=\"mn\" id=\"MathJax-Span-2325\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.031em, 1000.89em, 4.15em, -999.998em); top: -2.467em; left: 50%; margin-left: -0.417em;\"><span class=\"mtd\" id=\"MathJax-Span-2340\"><span class=\"mrow\" id=\"MathJax-Span-2341\"><span class=\"msubsup\" id=\"MathJax-Span-2342\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-2343\"><span class=\"mrow\" id=\"MathJax-Span-2344\"><span class=\"mi\" id=\"MathJax-Span-2345\" style=\"font-family: MathJax_Main;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.468em;\"><span class=\"mn\" id=\"MathJax-Span-2346\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-2347\" style=\"vertical-align: 2.239em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.277em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.514em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.768em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.023em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.522em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -2.051em; border-left: 0px solid; width: 0px; height: 4.705em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-59\">\\left|\\begin{array}{ccc}  \\mathrm{a} & \\mathrm{b} & \\mathrm{c} \\\\  \\mathrm{a}^2 & \\mathrm{b}^2 & \\mathrm{c}^2 \\\\  \\mathrm{a}^3 & \\mathrm{b}^3 & \\mathrm{c}^3  \\end{array}\\right|<\/script> = abc(a \u2013 b)(b \u2013 c)(c \u2013 a)<\/h3>\n\n\n\n<p>Answer:<br> LHS = abc<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-60-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-2348\" style=\"width: 6.479em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 5.594em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1005.45em, 6.293em, -999.998em); top: -4.471em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-2349\"><span class=\"mrow\" id=\"MathJax-Span-2350\"><span class=\"mo\" id=\"MathJax-Span-2351\" style=\"vertical-align: 2.192em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.417em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.349em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-2352\"><span style=\"display: inline-block; position: relative; width: 4.662em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.98em, 6.153em, -999.998em); top: -4.518em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-2353\"><span class=\"mrow\" id=\"MathJax-Span-2354\"><span class=\"mn\" id=\"MathJax-Span-2355\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.277em;\"><span class=\"mtd\" id=\"MathJax-Span-2362\"><span class=\"mrow\" id=\"MathJax-Span-2363\"><span class=\"mi\" id=\"MathJax-Span-2364\" style=\"font-family: MathJax_Math-italic;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.031em, 1000.98em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.464em;\"><span class=\"mtd\" id=\"MathJax-Span-2371\"><span class=\"mrow\" id=\"MathJax-Span-2372\"><span class=\"msubsup\" id=\"MathJax-Span-2373\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-2374\" style=\"font-family: MathJax_Math-italic;\">a<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-2375\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.522em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.84em, 6.153em, -999.998em); top: -4.518em; left: 1.959em;\"><span style=\"display: inline-block; position: relative; width: 0.841em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-2356\"><span class=\"mrow\" id=\"MathJax-Span-2357\"><span class=\"mn\" id=\"MathJax-Span-2358\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.42em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-2365\"><span class=\"mrow\" id=\"MathJax-Span-2366\"><span class=\"mi\" id=\"MathJax-Span-2367\" style=\"font-family: MathJax_Math-italic;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.031em, 1000.84em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.417em;\"><span class=\"mtd\" id=\"MathJax-Span-2376\"><span class=\"mrow\" id=\"MathJax-Span-2377\"><span class=\"msubsup\" id=\"MathJax-Span-2378\"><span style=\"display: inline-block; position: relative; width: 0.841em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-2379\" style=\"font-family: MathJax_Math-italic;\">b<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.422em;\"><span class=\"mn\" id=\"MathJax-Span-2380\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.522em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.84em, 6.153em, -999.998em); top: -4.518em; left: 3.823em;\"><span style=\"display: inline-block; position: relative; width: 0.841em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-2359\"><span class=\"mrow\" id=\"MathJax-Span-2360\"><span class=\"mn\" id=\"MathJax-Span-2361\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.42em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-2368\"><span class=\"mrow\" id=\"MathJax-Span-2369\"><span class=\"mi\" id=\"MathJax-Span-2370\" style=\"font-family: MathJax_Math-italic;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.031em, 1000.84em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.417em;\"><span class=\"mtd\" id=\"MathJax-Span-2381\"><span class=\"mrow\" id=\"MathJax-Span-2382\"><span class=\"msubsup\" id=\"MathJax-Span-2383\"><span style=\"display: inline-block; position: relative; width: 0.841em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-2384\" style=\"font-family: MathJax_Math-italic;\">c<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.422em;\"><span class=\"mn\" id=\"MathJax-Span-2385\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.522em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-2386\" style=\"vertical-align: 2.192em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.417em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.349em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.997em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-60\">\\left|\\begin{array}{ccc}  1 & 1 & 1 \\\\  a & b & c \\\\  a^2 & b^2 & c^2  \\end{array}\\right|<\/script><br> = abc<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-61-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-2387\" style=\"width: 11.419em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 9.835em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1009.7em, 6.293em, -999.998em); top: -4.471em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-2388\"><span class=\"mrow\" id=\"MathJax-Span-2389\"><span class=\"mo\" id=\"MathJax-Span-2390\" style=\"vertical-align: 2.192em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.417em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.349em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-2391\"><span style=\"display: inline-block; position: relative; width: 8.903em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.379em, 1003.08em, 6.247em, -999.998em); top: -4.564em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 3.078em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-2392\"><span class=\"mrow\" id=\"MathJax-Span-2393\"><span class=\"mn\" id=\"MathJax-Span-2394\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1002.19em, 4.243em, -999.998em); top: -3.959em; left: 50%; margin-left: -1.116em;\"><span class=\"mtd\" id=\"MathJax-Span-2401\"><span class=\"mrow\" id=\"MathJax-Span-2402\"><span class=\"mi\" id=\"MathJax-Span-2403\" style=\"font-family: MathJax_Math-italic;\">a<\/span><span class=\"mo\" id=\"MathJax-Span-2404\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"mi\" id=\"MathJax-Span-2405\" style=\"font-family: MathJax_Math-italic; padding-left: 0.235em;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.031em, 1003.08em, 4.243em, -999.998em); top: -2.561em; left: 50%; margin-left: -1.535em;\"><span class=\"mtd\" id=\"MathJax-Span-2414\"><span class=\"mrow\" id=\"MathJax-Span-2415\"><span class=\"msubsup\" id=\"MathJax-Span-2416\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-2417\" style=\"font-family: MathJax_Math-italic;\">a<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-2418\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-2419\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"msubsup\" id=\"MathJax-Span-2420\" style=\"padding-left: 0.235em;\"><span style=\"display: inline-block; position: relative; width: 0.841em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-2421\" style=\"font-family: MathJax_Math-italic;\">b<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.422em;\"><span class=\"mn\" id=\"MathJax-Span-2422\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.379em, 1002.99em, 6.247em, -999.998em); top: -4.564em; left: 4.056em;\"><span style=\"display: inline-block; position: relative; width: 2.985em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-2395\"><span class=\"mrow\" id=\"MathJax-Span-2396\"><span class=\"mn\" id=\"MathJax-Span-2397\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1002.1em, 4.243em, -999.998em); top: -3.959em; left: 50%; margin-left: -1.069em;\"><span class=\"mtd\" id=\"MathJax-Span-2406\"><span class=\"mrow\" id=\"MathJax-Span-2407\"><span class=\"mi\" id=\"MathJax-Span-2408\" style=\"font-family: MathJax_Math-italic;\">b<\/span><span class=\"mo\" id=\"MathJax-Span-2409\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"mi\" id=\"MathJax-Span-2410\" style=\"font-family: MathJax_Math-italic; padding-left: 0.235em;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.031em, 1002.99em, 4.243em, -999.998em); top: -2.561em; left: 50%; margin-left: -1.489em;\"><span class=\"mtd\" id=\"MathJax-Span-2423\"><span class=\"mrow\" id=\"MathJax-Span-2424\"><span class=\"msubsup\" id=\"MathJax-Span-2425\"><span style=\"display: inline-block; position: relative; width: 0.841em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-2426\" style=\"font-family: MathJax_Math-italic;\">b<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.422em;\"><span class=\"mn\" id=\"MathJax-Span-2427\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-2428\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"msubsup\" id=\"MathJax-Span-2429\" style=\"padding-left: 0.235em;\"><span style=\"display: inline-block; position: relative; width: 0.841em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-2430\" style=\"font-family: MathJax_Math-italic;\">c<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.422em;\"><span class=\"mn\" id=\"MathJax-Span-2431\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.84em, 6.153em, -999.998em); top: -4.518em; left: 8.017em;\"><span style=\"display: inline-block; position: relative; width: 0.841em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-2398\"><span class=\"mrow\" id=\"MathJax-Span-2399\"><span class=\"mn\" id=\"MathJax-Span-2400\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.42em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-2411\"><span class=\"mrow\" id=\"MathJax-Span-2412\"><span class=\"mi\" id=\"MathJax-Span-2413\" style=\"font-family: MathJax_Math-italic;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.031em, 1000.84em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.417em;\"><span class=\"mtd\" id=\"MathJax-Span-2432\"><span class=\"mrow\" id=\"MathJax-Span-2433\"><span class=\"msubsup\" id=\"MathJax-Span-2434\"><span style=\"display: inline-block; position: relative; width: 0.841em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-2435\" style=\"font-family: MathJax_Math-italic;\">c<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.422em;\"><span class=\"mn\" id=\"MathJax-Span-2436\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.522em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-2437\" style=\"vertical-align: 2.192em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.417em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.349em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.997em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-61\">\\left|\\begin{array}{ccc}  0 & 0 & 1 \\\\  a-b & b-c & c \\\\  a^2-b^2 & b^2-c^2 & c^2  \\end{array}\\right|<\/script> (Use operations C<sub>1<\/sub> \u2013 C<sub>2<\/sub>, C<sub>2<\/sub> \u2013 C<sub>3<\/sub>)<br> = abc [(a \u2013 b) (b<sup>2<\/sup> \u2013 c<sup>2<\/sup>) \u2013 (b \u2013 c) (a<sup>2<\/sup> \u2013 b<sup>2<\/sup>)]<br> = abc [(a \u2013 b) (b \u2013 c) (b + c) \u2013 (b \u2013 c) (a \u2013 b) (a + b)]<br> = abc (a \u2013 b) (b \u2013 c) [b + c \u2013 a \u2013 b]<br> = abc (a \u2013 b) (b \u2013 c) (c \u2013 a)<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Question 5.<br> Show that <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-62-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-2438\" style=\"width: 11.093em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 9.555em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1009.41em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-2439\"><span class=\"mrow\" id=\"MathJax-Span-2440\"><span class=\"mo\" id=\"MathJax-Span-2441\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-2442\"><span style=\"display: inline-block; position: relative; width: 8.623em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.425em, 1002.19em, 6.247em, -999.998em); top: -4.564em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 2.192em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.77em, 4.243em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.883em;\"><span class=\"mtd\" id=\"MathJax-Span-2443\"><span class=\"mrow\" id=\"MathJax-Span-2444\"><span class=\"mo\" id=\"MathJax-Span-2445\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-2446\" style=\"font-family: MathJax_Main;\">2<\/span><span class=\"mi\" id=\"MathJax-Span-2447\" style=\"font-family: MathJax_Math-italic;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1002.19em, 4.243em, -999.998em); top: -3.959em; left: 50%; margin-left: -1.116em;\"><span class=\"mtd\" id=\"MathJax-Span-2458\"><span class=\"mrow\" id=\"MathJax-Span-2459\"><span class=\"mi\" id=\"MathJax-Span-2460\" style=\"font-family: MathJax_Math-italic;\">a<\/span><span class=\"mo\" id=\"MathJax-Span-2461\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mi\" id=\"MathJax-Span-2462\" style=\"font-family: MathJax_Math-italic; padding-left: 0.235em;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.264em, 1002.19em, 4.243em, -999.998em); top: -2.561em; left: 50%; margin-left: -1.116em;\"><span class=\"mtd\" id=\"MathJax-Span-2473\"><span class=\"mrow\" id=\"MathJax-Span-2474\"><span class=\"mi\" id=\"MathJax-Span-2475\" style=\"font-family: MathJax_Math-italic;\">c<\/span><span class=\"mo\" id=\"MathJax-Span-2476\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mi\" id=\"MathJax-Span-2477\" style=\"font-family: MathJax_Math-italic; padding-left: 0.235em;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.379em, 1002.19em, 6.247em, -999.998em); top: -4.564em; left: 3.218em;\"><span style=\"display: inline-block; position: relative; width: 2.192em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1002.19em, 4.243em, -999.998em); top: -5.356em; left: 50%; margin-left: -1.116em;\"><span class=\"mtd\" id=\"MathJax-Span-2448\"><span class=\"mrow\" id=\"MathJax-Span-2449\"><span class=\"mi\" id=\"MathJax-Span-2450\" style=\"font-family: MathJax_Math-italic;\">a<\/span><span class=\"mo\" id=\"MathJax-Span-2451\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mi\" id=\"MathJax-Span-2452\" style=\"font-family: MathJax_Math-italic; padding-left: 0.235em;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1001.68em, 4.243em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.836em;\"><span class=\"mtd\" id=\"MathJax-Span-2463\"><span class=\"mrow\" id=\"MathJax-Span-2464\"><span class=\"mo\" id=\"MathJax-Span-2465\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-2466\" style=\"font-family: MathJax_Main;\">2<\/span><span class=\"mi\" id=\"MathJax-Span-2467\" style=\"font-family: MathJax_Math-italic;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1002.1em, 4.243em, -999.998em); top: -2.561em; left: 50%; margin-left: -1.069em;\"><span class=\"mtd\" id=\"MathJax-Span-2478\"><span class=\"mrow\" id=\"MathJax-Span-2479\"><span class=\"mi\" id=\"MathJax-Span-2480\" style=\"font-family: MathJax_Math-italic;\">c<\/span><span class=\"mo\" id=\"MathJax-Span-2481\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mi\" id=\"MathJax-Span-2482\" style=\"font-family: MathJax_Math-italic; padding-left: 0.235em;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.425em, 1002.19em, 6.153em, -999.998em); top: -4.471em; left: 6.433em;\"><span style=\"display: inline-block; position: relative; width: 2.192em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.264em, 1002.19em, 4.243em, -999.998em); top: -5.356em; left: 50%; margin-left: -1.116em;\"><span class=\"mtd\" id=\"MathJax-Span-2453\"><span class=\"mrow\" id=\"MathJax-Span-2454\"><span class=\"mi\" id=\"MathJax-Span-2455\" style=\"font-family: MathJax_Math-italic;\">c<\/span><span class=\"mo\" id=\"MathJax-Span-2456\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mi\" id=\"MathJax-Span-2457\" style=\"font-family: MathJax_Math-italic; padding-left: 0.235em;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1002.1em, 4.243em, -999.998em); top: -3.959em; left: 50%; margin-left: -1.069em;\"><span class=\"mtd\" id=\"MathJax-Span-2468\"><span class=\"mrow\" id=\"MathJax-Span-2469\"><span class=\"mi\" id=\"MathJax-Span-2470\" style=\"font-family: MathJax_Math-italic;\">b<\/span><span class=\"mo\" id=\"MathJax-Span-2471\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mi\" id=\"MathJax-Span-2472\" style=\"font-family: MathJax_Math-italic; padding-left: 0.235em;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.73em, 4.243em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.836em;\"><span class=\"mtd\" id=\"MathJax-Span-2483\"><span class=\"mrow\" id=\"MathJax-Span-2484\"><span class=\"mo\" id=\"MathJax-Span-2485\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-2486\" style=\"font-family: MathJax_Main;\">2<\/span><span class=\"mi\" id=\"MathJax-Span-2487\" style=\"font-family: MathJax_Math-italic;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-2488\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-62\">\\left|\\begin{array}{ccc}  -2 a & a+b & c+a \\\\  a+b & -2 b & b+c \\\\  c+a & c+b & -2 c  \\end{array}\\right|<\/script> = 4(a + b)(b + c)(c + a)<\/h3>\n\n\n\n<p>Answer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-5959\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3d-12.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(d) 12\" width=\"274\" height=\"421\" sizes=\"auto, (max-width: 274px) 100vw, 274px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(d) 12\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(d)\"><br> = 0 (\u2235 R<sub>1<\/sub> &amp; R<sub>3<\/sub> are similar)<br> \u2234 (a + b) is a factor of \u0394.<br> Similarly putting b + c = 0 and c + a = 0 we shall find that b + c and c + a are also factors of \u0394.<br> \u2235 \u0394 is a 3rd degree expression in a, b, c.<\/p>\n\n\n\n<p>Let \u0394 = k (a + b) (b + c) (c + a)<br> Where k \u2260 0 is a scalar.<br> Put a = 1, b = 1, c = 1 then<br> = k(1 + 1) (1 + 1) (1 + 1)<br> = 8k -2(4 \u2013 4) \u2013 2(-4 \u2013 4) + 2(4 + 4)<br> = 8k<br> \u21d2 -16 + 16 = 8k \u21d2 k = 4<br> \u0394 = 4(a + b) (b + c) (c + a)<br> Here <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-63-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-2489\" style=\"width: 11.093em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 9.555em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1009.41em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-2490\"><span class=\"mrow\" id=\"MathJax-Span-2491\"><span class=\"mo\" id=\"MathJax-Span-2492\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-2493\"><span style=\"display: inline-block; position: relative; width: 8.623em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.425em, 1002.19em, 6.247em, -999.998em); top: -4.564em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 2.192em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.77em, 4.243em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.883em;\"><span class=\"mtd\" id=\"MathJax-Span-2494\"><span class=\"mrow\" id=\"MathJax-Span-2495\"><span class=\"mo\" id=\"MathJax-Span-2496\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-2497\" style=\"font-family: MathJax_Main;\">2<\/span><span class=\"mi\" id=\"MathJax-Span-2498\" style=\"font-family: MathJax_Math-italic;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1002.19em, 4.243em, -999.998em); top: -3.959em; left: 50%; margin-left: -1.116em;\"><span class=\"mtd\" id=\"MathJax-Span-2509\"><span class=\"mrow\" id=\"MathJax-Span-2510\"><span class=\"mi\" id=\"MathJax-Span-2511\" style=\"font-family: MathJax_Math-italic;\">a<\/span><span class=\"mo\" id=\"MathJax-Span-2512\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mi\" id=\"MathJax-Span-2513\" style=\"font-family: MathJax_Math-italic; padding-left: 0.235em;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.264em, 1002.19em, 4.243em, -999.998em); top: -2.561em; left: 50%; margin-left: -1.116em;\"><span class=\"mtd\" id=\"MathJax-Span-2524\"><span class=\"mrow\" id=\"MathJax-Span-2525\"><span class=\"mi\" id=\"MathJax-Span-2526\" style=\"font-family: MathJax_Math-italic;\">c<\/span><span class=\"mo\" id=\"MathJax-Span-2527\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mi\" id=\"MathJax-Span-2528\" style=\"font-family: MathJax_Math-italic; padding-left: 0.235em;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.379em, 1002.19em, 6.247em, -999.998em); top: -4.564em; left: 3.218em;\"><span style=\"display: inline-block; position: relative; width: 2.192em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1002.19em, 4.243em, -999.998em); top: -5.356em; left: 50%; margin-left: -1.116em;\"><span class=\"mtd\" id=\"MathJax-Span-2499\"><span class=\"mrow\" id=\"MathJax-Span-2500\"><span class=\"mi\" id=\"MathJax-Span-2501\" style=\"font-family: MathJax_Math-italic;\">a<\/span><span class=\"mo\" id=\"MathJax-Span-2502\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mi\" id=\"MathJax-Span-2503\" style=\"font-family: MathJax_Math-italic; padding-left: 0.235em;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1001.68em, 4.243em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.836em;\"><span class=\"mtd\" id=\"MathJax-Span-2514\"><span class=\"mrow\" id=\"MathJax-Span-2515\"><span class=\"mo\" id=\"MathJax-Span-2516\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-2517\" style=\"font-family: MathJax_Main;\">2<\/span><span class=\"mi\" id=\"MathJax-Span-2518\" style=\"font-family: MathJax_Math-italic;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1002.1em, 4.243em, -999.998em); top: -2.561em; left: 50%; margin-left: -1.069em;\"><span class=\"mtd\" id=\"MathJax-Span-2529\"><span class=\"mrow\" id=\"MathJax-Span-2530\"><span class=\"mi\" id=\"MathJax-Span-2531\" style=\"font-family: MathJax_Math-italic;\">c<\/span><span class=\"mo\" id=\"MathJax-Span-2532\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mi\" id=\"MathJax-Span-2533\" style=\"font-family: MathJax_Math-italic; padding-left: 0.235em;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.425em, 1002.19em, 6.153em, -999.998em); top: -4.471em; left: 6.433em;\"><span style=\"display: inline-block; position: relative; width: 2.192em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.264em, 1002.19em, 4.243em, -999.998em); top: -5.356em; left: 50%; margin-left: -1.116em;\"><span class=\"mtd\" id=\"MathJax-Span-2504\"><span class=\"mrow\" id=\"MathJax-Span-2505\"><span class=\"mi\" id=\"MathJax-Span-2506\" style=\"font-family: MathJax_Math-italic;\">c<\/span><span class=\"mo\" id=\"MathJax-Span-2507\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mi\" id=\"MathJax-Span-2508\" style=\"font-family: MathJax_Math-italic; padding-left: 0.235em;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1002.1em, 4.243em, -999.998em); top: -3.959em; left: 50%; margin-left: -1.069em;\"><span class=\"mtd\" id=\"MathJax-Span-2519\"><span class=\"mrow\" id=\"MathJax-Span-2520\"><span class=\"mi\" id=\"MathJax-Span-2521\" style=\"font-family: MathJax_Math-italic;\">b<\/span><span class=\"mo\" id=\"MathJax-Span-2522\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mi\" id=\"MathJax-Span-2523\" style=\"font-family: MathJax_Math-italic; padding-left: 0.235em;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.73em, 4.243em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.836em;\"><span class=\"mtd\" id=\"MathJax-Span-2534\"><span class=\"mrow\" id=\"MathJax-Span-2535\"><span class=\"mo\" id=\"MathJax-Span-2536\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-2537\" style=\"font-family: MathJax_Main;\">2<\/span><span class=\"mi\" id=\"MathJax-Span-2538\" style=\"font-family: MathJax_Math-italic;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-2539\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-63\">\\left|\\begin{array}{ccc}  -2 a & a+b & c+a \\\\  a+b & -2 b & b+c \\\\  c+a & c+b & -2 c  \\end{array}\\right|<\/script> = 4(a + b)(b + c)(c + a)<\/p>\n\n\n\n<p>Question 6.<br> Show that <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-64-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-2540\" style=\"width: 11.419em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 9.835em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1009.7em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-2541\"><span class=\"mrow\" id=\"MathJax-Span-2542\"><span class=\"mo\" id=\"MathJax-Span-2543\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-2544\"><span style=\"display: inline-block; position: relative; width: 8.903em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.379em, 1002.29em, 6.247em, -999.998em); top: -4.564em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 2.286em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1002.29em, 4.243em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-2545\"><span class=\"mrow\" id=\"MathJax-Span-2546\"><span class=\"texatom\" id=\"MathJax-Span-2547\"><span class=\"mrow\" id=\"MathJax-Span-2548\"><span class=\"mi\" id=\"MathJax-Span-2549\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-2550\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"texatom\" id=\"MathJax-Span-2551\" style=\"padding-left: 0.235em;\"><span class=\"mrow\" id=\"MathJax-Span-2552\"><span class=\"mi\" id=\"MathJax-Span-2553\" style=\"font-family: MathJax_Main;\">b<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1002.24em, 4.243em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-2572\"><span class=\"mrow\" id=\"MathJax-Span-2573\"><span class=\"texatom\" id=\"MathJax-Span-2574\"><span class=\"mrow\" id=\"MathJax-Span-2575\"><span class=\"mi\" id=\"MathJax-Span-2576\" style=\"font-family: MathJax_Main;\">b<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-2577\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"texatom\" id=\"MathJax-Span-2578\" style=\"padding-left: 0.235em;\"><span class=\"mrow\" id=\"MathJax-Span-2579\"><span class=\"mi\" id=\"MathJax-Span-2580\" style=\"font-family: MathJax_Main;\">c<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.264em, 1002.19em, 4.243em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-2599\"><span class=\"mrow\" id=\"MathJax-Span-2600\"><span class=\"texatom\" id=\"MathJax-Span-2601\"><span class=\"mrow\" id=\"MathJax-Span-2602\"><span class=\"mi\" id=\"MathJax-Span-2603\" style=\"font-family: MathJax_Main;\">c<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-2604\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"texatom\" id=\"MathJax-Span-2605\" style=\"padding-left: 0.235em;\"><span class=\"mrow\" id=\"MathJax-Span-2606\"><span class=\"mi\" id=\"MathJax-Span-2607\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.379em, 1002.29em, 6.247em, -999.998em); top: -4.564em; left: 3.311em;\"><span style=\"display: inline-block; position: relative; width: 2.286em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1002.24em, 4.243em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-2554\"><span class=\"mrow\" id=\"MathJax-Span-2555\"><span class=\"texatom\" id=\"MathJax-Span-2556\"><span class=\"mrow\" id=\"MathJax-Span-2557\"><span class=\"mi\" id=\"MathJax-Span-2558\" style=\"font-family: MathJax_Main;\">b<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-2559\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"texatom\" id=\"MathJax-Span-2560\" style=\"padding-left: 0.235em;\"><span class=\"mrow\" id=\"MathJax-Span-2561\"><span class=\"mi\" id=\"MathJax-Span-2562\" style=\"font-family: MathJax_Main;\">c<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.264em, 1002.19em, 4.243em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-2581\"><span class=\"mrow\" id=\"MathJax-Span-2582\"><span class=\"texatom\" id=\"MathJax-Span-2583\"><span class=\"mrow\" id=\"MathJax-Span-2584\"><span class=\"mi\" id=\"MathJax-Span-2585\" style=\"font-family: MathJax_Main;\">c<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-2586\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"texatom\" id=\"MathJax-Span-2587\" style=\"padding-left: 0.235em;\"><span class=\"mrow\" id=\"MathJax-Span-2588\"><span class=\"mi\" id=\"MathJax-Span-2589\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1002.29em, 4.243em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-2608\"><span class=\"mrow\" id=\"MathJax-Span-2609\"><span class=\"texatom\" id=\"MathJax-Span-2610\"><span class=\"mrow\" id=\"MathJax-Span-2611\"><span class=\"mi\" id=\"MathJax-Span-2612\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-2613\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"texatom\" id=\"MathJax-Span-2614\" style=\"padding-left: 0.235em;\"><span class=\"mrow\" id=\"MathJax-Span-2615\"><span class=\"mi\" id=\"MathJax-Span-2616\" style=\"font-family: MathJax_Main;\">b<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.425em, 1002.29em, 6.153em, -999.998em); top: -4.471em; left: 6.619em;\"><span style=\"display: inline-block; position: relative; width: 2.286em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.264em, 1002.19em, 4.243em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-2563\"><span class=\"mrow\" id=\"MathJax-Span-2564\"><span class=\"texatom\" id=\"MathJax-Span-2565\"><span class=\"mrow\" id=\"MathJax-Span-2566\"><span class=\"mi\" id=\"MathJax-Span-2567\" style=\"font-family: MathJax_Main;\">c<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-2568\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"texatom\" id=\"MathJax-Span-2569\" style=\"padding-left: 0.235em;\"><span class=\"mrow\" id=\"MathJax-Span-2570\"><span class=\"mi\" id=\"MathJax-Span-2571\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1002.29em, 4.243em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-2590\"><span class=\"mrow\" id=\"MathJax-Span-2591\"><span class=\"texatom\" id=\"MathJax-Span-2592\"><span class=\"mrow\" id=\"MathJax-Span-2593\"><span class=\"mi\" id=\"MathJax-Span-2594\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-2595\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"texatom\" id=\"MathJax-Span-2596\" style=\"padding-left: 0.235em;\"><span class=\"mrow\" id=\"MathJax-Span-2597\"><span class=\"mi\" id=\"MathJax-Span-2598\" style=\"font-family: MathJax_Main;\">b<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1002.24em, 4.243em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-2617\"><span class=\"mrow\" id=\"MathJax-Span-2618\"><span class=\"texatom\" id=\"MathJax-Span-2619\"><span class=\"mrow\" id=\"MathJax-Span-2620\"><span class=\"mi\" id=\"MathJax-Span-2621\" style=\"font-family: MathJax_Main;\">b<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-2622\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"texatom\" id=\"MathJax-Span-2623\" style=\"padding-left: 0.235em;\"><span class=\"mrow\" id=\"MathJax-Span-2624\"><span class=\"mi\" id=\"MathJax-Span-2625\" style=\"font-family: MathJax_Main;\">c<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-2626\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-64\">\\left|\\begin{array}{lll}  \\mathrm{a}-\\mathrm{b} & \\mathrm{b}-\\mathrm{c} & \\mathrm{c}-\\mathrm{a} \\\\  \\mathrm{b}-\\mathrm{c} & \\mathrm{c}-\\mathrm{a} & \\mathrm{a}-\\mathrm{b} \\\\  \\mathrm{c}-\\mathrm{a} & \\mathrm{a}-\\mathrm{b} & \\mathrm{b}-\\mathrm{c}  \\end{array}\\right|<\/script> = 0<br> Answer:<br> R<sub>1<\/sub> + (R<sub>2<\/sub> + R<sub>3<\/sub>) given<br> <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-65-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-2627\" style=\"width: 11.093em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 9.555em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1009.41em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-2628\"><span class=\"mrow\" id=\"MathJax-Span-2629\"><span class=\"mo\" id=\"MathJax-Span-2630\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-2631\"><span style=\"display: inline-block; position: relative; width: 8.623em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.425em, 1002.19em, 6.247em, -999.998em); top: -4.564em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 2.192em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-2632\"><span class=\"mrow\" id=\"MathJax-Span-2633\"><span class=\"mn\" id=\"MathJax-Span-2634\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1002.1em, 4.243em, -999.998em); top: -3.959em; left: 50%; margin-left: -1.069em;\"><span class=\"mtd\" id=\"MathJax-Span-2641\"><span class=\"mrow\" id=\"MathJax-Span-2642\"><span class=\"mi\" id=\"MathJax-Span-2643\" style=\"font-family: MathJax_Math-italic;\">b<\/span><span class=\"mo\" id=\"MathJax-Span-2644\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"mi\" id=\"MathJax-Span-2645\" style=\"font-family: MathJax_Math-italic; padding-left: 0.235em;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.264em, 1002.19em, 4.243em, -999.998em); top: -2.561em; left: 50%; margin-left: -1.116em;\"><span class=\"mtd\" id=\"MathJax-Span-2656\"><span class=\"mrow\" id=\"MathJax-Span-2657\"><span class=\"mi\" id=\"MathJax-Span-2658\" style=\"font-family: MathJax_Math-italic;\">c<\/span><span class=\"mo\" id=\"MathJax-Span-2659\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"mi\" id=\"MathJax-Span-2660\" style=\"font-family: MathJax_Math-italic; padding-left: 0.235em;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.425em, 1002.19em, 6.247em, -999.998em); top: -4.564em; left: 3.218em;\"><span style=\"display: inline-block; position: relative; width: 2.192em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-2635\"><span class=\"mrow\" id=\"MathJax-Span-2636\"><span class=\"mn\" id=\"MathJax-Span-2637\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.264em, 1002.19em, 4.243em, -999.998em); top: -3.959em; left: 50%; margin-left: -1.116em;\"><span class=\"mtd\" id=\"MathJax-Span-2646\"><span class=\"mrow\" id=\"MathJax-Span-2647\"><span class=\"mi\" id=\"MathJax-Span-2648\" style=\"font-family: MathJax_Math-italic;\">c<\/span><span class=\"mo\" id=\"MathJax-Span-2649\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"mi\" id=\"MathJax-Span-2650\" style=\"font-family: MathJax_Math-italic; padding-left: 0.235em;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1002.19em, 4.243em, -999.998em); top: -2.561em; left: 50%; margin-left: -1.116em;\"><span class=\"mtd\" id=\"MathJax-Span-2661\"><span class=\"mrow\" id=\"MathJax-Span-2662\"><span class=\"mi\" id=\"MathJax-Span-2663\" style=\"font-family: MathJax_Math-italic;\">a<\/span><span class=\"mo\" id=\"MathJax-Span-2664\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"mi\" id=\"MathJax-Span-2665\" style=\"font-family: MathJax_Math-italic; padding-left: 0.235em;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.425em, 1002.19em, 6.247em, -999.998em); top: -4.564em; left: 6.433em;\"><span style=\"display: inline-block; position: relative; width: 2.192em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-2638\"><span class=\"mrow\" id=\"MathJax-Span-2639\"><span class=\"mn\" id=\"MathJax-Span-2640\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1002.19em, 4.243em, -999.998em); top: -3.959em; left: 50%; margin-left: -1.116em;\"><span class=\"mtd\" id=\"MathJax-Span-2651\"><span class=\"mrow\" id=\"MathJax-Span-2652\"><span class=\"mi\" id=\"MathJax-Span-2653\" style=\"font-family: MathJax_Math-italic;\">a<\/span><span class=\"mo\" id=\"MathJax-Span-2654\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"mi\" id=\"MathJax-Span-2655\" style=\"font-family: MathJax_Math-italic; padding-left: 0.235em;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1002.1em, 4.243em, -999.998em); top: -2.561em; left: 50%; margin-left: -1.069em;\"><span class=\"mtd\" id=\"MathJax-Span-2666\"><span class=\"mrow\" id=\"MathJax-Span-2667\"><span class=\"mi\" id=\"MathJax-Span-2668\" style=\"font-family: MathJax_Math-italic;\">b<\/span><span class=\"mo\" id=\"MathJax-Span-2669\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"mi\" id=\"MathJax-Span-2670\" style=\"font-family: MathJax_Math-italic; padding-left: 0.235em;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-2671\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-65\">\\left|\\begin{array}{ccc}  0 & 0 & 0 \\\\  b-c & c-a & a-b \\\\  c-a & a-b & b-c  \\end{array}\\right|<\/script><br> = 0 (\u2235 If one row or column elements of a square matrix are zeroes then the value of the determinant of that matrix is equal to zero)<br> = RHS.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Question 7.<br> Show that <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-66-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-2672\" style=\"width: 8.157em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 7.039em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1006.9em, 6.293em, -999.998em); top: -4.471em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-2673\"><span class=\"mrow\" id=\"MathJax-Span-2674\"><span class=\"mo\" id=\"MathJax-Span-2675\" style=\"vertical-align: 2.192em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.37em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.514em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.815em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.069em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-2676\"><span style=\"display: inline-block; position: relative; width: 6.107em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.42em, 6.153em, -999.998em); top: -4.518em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-2677\"><span class=\"mrow\" id=\"MathJax-Span-2678\"><span class=\"mn\" id=\"MathJax-Span-2679\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-2691\"><span class=\"mrow\" id=\"MathJax-Span-2692\"><span class=\"mn\" id=\"MathJax-Span-2693\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.514em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-2705\"><span class=\"mrow\" id=\"MathJax-Span-2706\"><span class=\"mn\" id=\"MathJax-Span-2707\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.522em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.379em, 1000.51em, 5.967em, -999.998em); top: -4.331em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-2680\"><span class=\"mrow\" id=\"MathJax-Span-2681\"><span class=\"mi\" id=\"MathJax-Span-2682\" style=\"font-family: MathJax_Math-italic;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.42em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-2694\"><span class=\"mrow\" id=\"MathJax-Span-2695\"><span class=\"mi\" id=\"MathJax-Span-2696\" style=\"font-family: MathJax_Math-italic;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.42em, 4.15em, -999.998em); top: -2.514em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-2708\"><span class=\"mrow\" id=\"MathJax-Span-2709\"><span class=\"mi\" id=\"MathJax-Span-2710\" style=\"font-family: MathJax_Math-italic;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.336em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.472em, 1003.08em, 6.526em, -999.998em); top: -4.797em; left: 3.031em;\"><span style=\"display: inline-block; position: relative; width: 3.078em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.031em, 1003.08em, 4.243em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-2683\"><span class=\"mrow\" id=\"MathJax-Span-2684\"><span class=\"msubsup\" id=\"MathJax-Span-2685\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-2686\" style=\"font-family: MathJax_Math-italic;\">a<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-2687\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-2688\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"mi\" id=\"MathJax-Span-2689\" style=\"font-family: MathJax_Math-italic; padding-left: 0.235em;\">b<\/span><span class=\"mi\" id=\"MathJax-Span-2690\" style=\"font-family: MathJax_Math-italic;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.031em, 1003.03em, 4.243em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-2697\"><span class=\"mrow\" id=\"MathJax-Span-2698\"><span class=\"msubsup\" id=\"MathJax-Span-2699\"><span style=\"display: inline-block; position: relative; width: 0.841em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-2700\" style=\"font-family: MathJax_Math-italic;\">b<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.422em;\"><span class=\"mn\" id=\"MathJax-Span-2701\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-2702\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"mi\" id=\"MathJax-Span-2703\" style=\"font-family: MathJax_Math-italic; padding-left: 0.235em;\">c<\/span><span class=\"mi\" id=\"MathJax-Span-2704\" style=\"font-family: MathJax_Math-italic;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.031em, 1003.08em, 4.243em, -999.998em); top: -2.514em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-2711\"><span class=\"mrow\" id=\"MathJax-Span-2712\"><span class=\"msubsup\" id=\"MathJax-Span-2713\"><span style=\"display: inline-block; position: relative; width: 0.841em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-2714\" style=\"font-family: MathJax_Math-italic;\">c<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.422em;\"><span class=\"mn\" id=\"MathJax-Span-2715\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-2716\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"mi\" id=\"MathJax-Span-2717\" style=\"font-family: MathJax_Math-italic; padding-left: 0.235em;\">a<\/span><span class=\"mi\" id=\"MathJax-Span-2718\" style=\"font-family: MathJax_Math-italic;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.802em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-2719\" style=\"vertical-align: 2.192em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.37em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.514em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.815em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.069em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.997em; border-left: 0px solid; width: 0px; height: 4.651em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-66\">\\left|\\begin{array}{lll}  1 & a & a^2-b c \\\\  1 & b & b^2-c a \\\\  1 & c & c^2-a b  \\end{array}\\right|<\/script> = 0<\/h3>\n\n\n\n<p>Answer:<br> Make operations R<sub>2<\/sub> \u2013 R<sub>1<\/sub>, R<sub>3<\/sub> \u2013 R<sub>1<\/sub> then the given determinant.<br>\n\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-5958\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3d-13.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(d) 13\" width=\"276\" height=\"343\" sizes=\"auto, (max-width: 276px) 100vw, 276px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(d) 13\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(d)\"><\/p>\n\n\n\n<p>Question 8.<br> Show that <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-67-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-2720\" style=\"width: 5.408em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.662em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1004.52em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-2721\"><span class=\"mrow\" id=\"MathJax-Span-2722\"><span class=\"mo\" id=\"MathJax-Span-2723\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-2724\"><span style=\"display: inline-block; position: relative; width: 3.73em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.51em, 5.827em, -999.998em); top: -4.238em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.562em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-2725\"><span class=\"mrow\" id=\"MathJax-Span-2726\"><span class=\"mi\" id=\"MathJax-Span-2727\" style=\"font-family: MathJax_Math-italic;\">x<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-2734\"><span class=\"mrow\" id=\"MathJax-Span-2735\"><span class=\"mi\" id=\"MathJax-Span-2736\" style=\"font-family: MathJax_Math-italic;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.15em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-2743\"><span class=\"mrow\" id=\"MathJax-Span-2744\"><span class=\"mi\" id=\"MathJax-Span-2745\" style=\"font-family: MathJax_Math-italic;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.243em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.51em, 5.827em, -999.998em); top: -4.238em; left: 1.587em;\"><span style=\"display: inline-block; position: relative; width: 0.562em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-2728\"><span class=\"mrow\" id=\"MathJax-Span-2729\"><span class=\"mi\" id=\"MathJax-Span-2730\" style=\"font-family: MathJax_Math-italic;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-2737\"><span class=\"mrow\" id=\"MathJax-Span-2738\"><span class=\"mi\" id=\"MathJax-Span-2739\" style=\"font-family: MathJax_Math-italic;\">x<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.15em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-2746\"><span class=\"mrow\" id=\"MathJax-Span-2747\"><span class=\"mi\" id=\"MathJax-Span-2748\" style=\"font-family: MathJax_Math-italic;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.243em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.51em, 5.827em, -999.998em); top: -4.238em; left: 3.124em;\"><span style=\"display: inline-block; position: relative; width: 0.562em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-2731\"><span class=\"mrow\" id=\"MathJax-Span-2732\"><span class=\"mi\" id=\"MathJax-Span-2733\" style=\"font-family: MathJax_Math-italic;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-2740\"><span class=\"mrow\" id=\"MathJax-Span-2741\"><span class=\"mi\" id=\"MathJax-Span-2742\" style=\"font-family: MathJax_Math-italic;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.15em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-2749\"><span class=\"mrow\" id=\"MathJax-Span-2750\"><span class=\"mi\" id=\"MathJax-Span-2751\" style=\"font-family: MathJax_Math-italic;\">x<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.243em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-2752\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-67\">\\left|\\begin{array}{lll}  x & a & a \\\\  a & x & a \\\\  a & a & x  \\end{array}\\right|<\/script> = (x + 2a)(x \u2013 a)<sup>2<\/sup>.\n\n<\/p>\n\n\n\n<p>Answer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-5957\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3d-14.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(d) 14\" width=\"311\" height=\"406\" sizes=\"auto, (max-width: 311px) 100vw, 311px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(d) 14\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(d)\"><\/p>\n","protected":false},"excerpt":{"rendered":"<p>I. Question 1. Find the determinants of the following matrices. (i) [211\u22125] Answer: Let A = [211\u22125] then determinant A = det A = |A| = 2(-5) \u2013 1(1) = -10 \u2013 1 = -11 (ii) [4\u2212652] Answer: Let A = [4\u2212652] then det A = 4(2) \u2013 5(-6) = 8 + 30 = 38 [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":2699,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[1544,15,38],"tags":[1615,1574,1612,1616,1611,1610,1609,1613,1614],"class_list":{"0":"post-2694","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-study-material-tg-inter","8":"category-telangana","9":"category-tg-inter","10":"tag-bie-telangana-maths-matrices-solutions","11":"tag-inter-1st-year-maths-1a-matrices-important-questions","12":"tag-telangana-intermediate-maths-1a-chapter-3-solutions","13":"tag-ts-inter-1a-maths-textbook-solutions","14":"tag-ts-inter-1st-year-maths-1a-matrices-solutions","15":"tag-ts-inter-1st-year-matrices-solutions-pdf","16":"tag-ts-inter-maths-1a-exercise-3-solutions","17":"tag-ts-inter-maths-1a-matrices-exercise-3-answers","18":"tag-ts-intermediate-1st-year-maths-1a-guide"},"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\r\n<title>TS Inter 1st Year Maths 1A Matrices Solutions Exercise 3(d)<\/title>\r\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\r\n<link rel=\"canonical\" href=\"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3d\/\" \/>\r\n<meta property=\"og:locale\" content=\"en_US\" \/>\r\n<meta property=\"og:type\" content=\"article\" \/>\r\n<meta property=\"og:title\" content=\"TS Inter 1st Year Maths 1A Matrices Solutions Exercise 3(d)\" \/>\r\n<meta property=\"og:description\" content=\"I. Question 1. Find the determinants of the following matrices. (i) [211\u22125] Answer: Let A = [211\u22125] then determinant A = det A = |A| = 2(-5) \u2013 1(1) = -10 \u2013 1 = -11 (ii) [4\u2212652] Answer: Let A = [4\u2212652] then det A = 4(2) \u2013 5(-6) = 8 + 30 = 38 [&hellip;]\" \/>\r\n<meta property=\"og:url\" content=\"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3d\/\" \/>\r\n<meta property=\"og:site_name\" content=\"Manabadi Boards\" \/>\r\n<meta property=\"article:published_time\" content=\"2026-01-08T08:23:31+00:00\" \/>\r\n<meta property=\"article:modified_time\" content=\"2026-01-08T11:05:53+00:00\" \/>\r\n<meta property=\"og:image\" content=\"https:\/\/boardscdn.manabadi.co.in\/wp-content\/uploads\/2026\/01\/07114619\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3d.jpg\" \/>\r\n\t<meta property=\"og:image:width\" content=\"900\" \/>\r\n\t<meta property=\"og:image:height\" content=\"600\" \/>\r\n\t<meta property=\"og:image:type\" content=\"image\/jpeg\" \/>\r\n<meta name=\"author\" content=\"Junaid\" \/>\r\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\r\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Junaid\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"14 minutes\" \/>\r\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3d\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3d\\\/\"},\"author\":{\"name\":\"Junaid\",\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/#\\\/schema\\\/person\\\/a55695e135e9dc26536b290e31b4179d\"},\"headline\":\"TS Inter 1st Year Maths 1A Matrices Solutions Exercise 3(d)\",\"datePublished\":\"2026-01-08T08:23:31+00:00\",\"dateModified\":\"2026-01-08T11:05:53+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3d\\\/\"},\"wordCount\":1055,\"commentCount\":0,\"image\":{\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3d\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/boardscdn.manabadi.co.in\\\/wp-content\\\/uploads\\\/2026\\\/01\\\/07114619\\\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3d.jpg\",\"keywords\":[\"BIE Telangana Maths Matrices solutions\",\"Inter 1st Year Maths 1A Matrices Important Questions\",\"Telangana Intermediate Maths 1A Chapter 3 Solutions\",\"TS Inter 1A Maths textbook solutions\",\"TS Inter 1st Year Maths 1A Matrices Solutions\",\"TS Inter 1st Year Matrices Solutions PDF\",\"TS Inter Maths 1A Exercise 3 Solutions\",\"TS Inter Maths 1A Matrices Exercise 3 answers\",\"TS Intermediate 1st year Maths 1A Guide\"],\"articleSection\":[\"Study Material\",\"Telangana\",\"TG Inter\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3d\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3d\\\/\",\"url\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3d\\\/\",\"name\":\"TS Inter 1st Year Maths 1A Matrices Solutions Exercise 3(d)\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3d\\\/#primaryimage\"},\"image\":{\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3d\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/boardscdn.manabadi.co.in\\\/wp-content\\\/uploads\\\/2026\\\/01\\\/07114619\\\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3d.jpg\",\"datePublished\":\"2026-01-08T08:23:31+00:00\",\"dateModified\":\"2026-01-08T11:05:53+00:00\",\"author\":{\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/#\\\/schema\\\/person\\\/a55695e135e9dc26536b290e31b4179d\"},\"breadcrumb\":{\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3d\\\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3d\\\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3d\\\/#primaryimage\",\"url\":\"https:\\\/\\\/boardscdn.manabadi.co.in\\\/wp-content\\\/uploads\\\/2026\\\/01\\\/07114619\\\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3d.jpg\",\"contentUrl\":\"https:\\\/\\\/boardscdn.manabadi.co.in\\\/wp-content\\\/uploads\\\/2026\\\/01\\\/07114619\\\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3d.jpg\",\"width\":900,\"height\":600,\"caption\":\"TS Inter 1st Year Maths 1A Matrices Solutions Exercise\"},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3d\\\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"TS Inter 1st Year Maths 1A Matrices Solutions Exercise 3(d)\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/#website\",\"url\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/\",\"name\":\"Manabadi Boards\",\"description\":\"\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"en-US\"},{\"@type\":\"Person\",\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/#\\\/schema\\\/person\\\/a55695e135e9dc26536b290e31b4179d\",\"name\":\"Junaid\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\\\/\\\/secure.gravatar.com\\\/avatar\\\/cf93d7547c6d8b1489ad346979c0e0d181d53fe095610d9d059830de5f332e0e?s=96&d=mm&r=g\",\"url\":\"https:\\\/\\\/secure.gravatar.com\\\/avatar\\\/cf93d7547c6d8b1489ad346979c0e0d181d53fe095610d9d059830de5f332e0e?s=96&d=mm&r=g\",\"contentUrl\":\"https:\\\/\\\/secure.gravatar.com\\\/avatar\\\/cf93d7547c6d8b1489ad346979c0e0d181d53fe095610d9d059830de5f332e0e?s=96&d=mm&r=g\",\"caption\":\"Junaid\"}}]}<\/script>\r\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"TS Inter 1st Year Maths 1A Matrices Solutions Exercise 3(d)","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3d\/","og_locale":"en_US","og_type":"article","og_title":"TS Inter 1st Year Maths 1A Matrices Solutions Exercise 3(d)","og_description":"I. Question 1. Find the determinants of the following matrices. (i) [211\u22125] Answer: Let A = [211\u22125] then determinant A = det A = |A| = 2(-5) \u2013 1(1) = -10 \u2013 1 = -11 (ii) [4\u2212652] Answer: Let A = [4\u2212652] then det A = 4(2) \u2013 5(-6) = 8 + 30 = 38 [&hellip;]","og_url":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3d\/","og_site_name":"Manabadi Boards","article_published_time":"2026-01-08T08:23:31+00:00","article_modified_time":"2026-01-08T11:05:53+00:00","og_image":[{"width":900,"height":600,"url":"https:\/\/boardscdn.manabadi.co.in\/wp-content\/uploads\/2026\/01\/07114619\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3d.jpg","type":"image\/jpeg"}],"author":"Junaid","twitter_card":"summary_large_image","twitter_misc":{"Written by":"Junaid","Est. reading time":"14 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3d\/#article","isPartOf":{"@id":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3d\/"},"author":{"name":"Junaid","@id":"https:\/\/www.manabadi.co.in\/boards\/#\/schema\/person\/a55695e135e9dc26536b290e31b4179d"},"headline":"TS Inter 1st Year Maths 1A Matrices Solutions Exercise 3(d)","datePublished":"2026-01-08T08:23:31+00:00","dateModified":"2026-01-08T11:05:53+00:00","mainEntityOfPage":{"@id":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3d\/"},"wordCount":1055,"commentCount":0,"image":{"@id":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3d\/#primaryimage"},"thumbnailUrl":"https:\/\/boardscdn.manabadi.co.in\/wp-content\/uploads\/2026\/01\/07114619\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3d.jpg","keywords":["BIE Telangana Maths Matrices solutions","Inter 1st Year Maths 1A Matrices Important Questions","Telangana Intermediate Maths 1A Chapter 3 Solutions","TS Inter 1A Maths textbook solutions","TS Inter 1st Year Maths 1A Matrices Solutions","TS Inter 1st Year Matrices Solutions PDF","TS Inter Maths 1A Exercise 3 Solutions","TS Inter Maths 1A Matrices Exercise 3 answers","TS Intermediate 1st year Maths 1A Guide"],"articleSection":["Study Material","Telangana","TG Inter"],"inLanguage":"en-US","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3d\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3d\/","url":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3d\/","name":"TS Inter 1st Year Maths 1A Matrices Solutions Exercise 3(d)","isPartOf":{"@id":"https:\/\/www.manabadi.co.in\/boards\/#website"},"primaryImageOfPage":{"@id":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3d\/#primaryimage"},"image":{"@id":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3d\/#primaryimage"},"thumbnailUrl":"https:\/\/boardscdn.manabadi.co.in\/wp-content\/uploads\/2026\/01\/07114619\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3d.jpg","datePublished":"2026-01-08T08:23:31+00:00","dateModified":"2026-01-08T11:05:53+00:00","author":{"@id":"https:\/\/www.manabadi.co.in\/boards\/#\/schema\/person\/a55695e135e9dc26536b290e31b4179d"},"breadcrumb":{"@id":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3d\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3d\/"]}]},{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3d\/#primaryimage","url":"https:\/\/boardscdn.manabadi.co.in\/wp-content\/uploads\/2026\/01\/07114619\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3d.jpg","contentUrl":"https:\/\/boardscdn.manabadi.co.in\/wp-content\/uploads\/2026\/01\/07114619\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3d.jpg","width":900,"height":600,"caption":"TS Inter 1st Year Maths 1A Matrices Solutions Exercise"},{"@type":"BreadcrumbList","@id":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3d\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/www.manabadi.co.in\/boards\/"},{"@type":"ListItem","position":2,"name":"TS Inter 1st Year Maths 1A Matrices Solutions Exercise 3(d)"}]},{"@type":"WebSite","@id":"https:\/\/www.manabadi.co.in\/boards\/#website","url":"https:\/\/www.manabadi.co.in\/boards\/","name":"Manabadi Boards","description":"","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/www.manabadi.co.in\/boards\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-US"},{"@type":"Person","@id":"https:\/\/www.manabadi.co.in\/boards\/#\/schema\/person\/a55695e135e9dc26536b290e31b4179d","name":"Junaid","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/secure.gravatar.com\/avatar\/cf93d7547c6d8b1489ad346979c0e0d181d53fe095610d9d059830de5f332e0e?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/cf93d7547c6d8b1489ad346979c0e0d181d53fe095610d9d059830de5f332e0e?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/cf93d7547c6d8b1489ad346979c0e0d181d53fe095610d9d059830de5f332e0e?s=96&d=mm&r=g","caption":"Junaid"}}]}},"_links":{"self":[{"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/posts\/2694","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/comments?post=2694"}],"version-history":[{"count":3,"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/posts\/2694\/revisions"}],"predecessor-version":[{"id":2714,"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/posts\/2694\/revisions\/2714"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/media\/2699"}],"wp:attachment":[{"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/media?parent=2694"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/categories?post=2694"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/tags?post=2694"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}