{"id":2786,"date":"2026-01-12T11:14:32","date_gmt":"2026-01-12T05:44:32","guid":{"rendered":"https:\/\/www.manabadi.co.in\/boards\/?p=2786"},"modified":"2026-01-12T14:15:43","modified_gmt":"2026-01-12T08:45:43","slug":"ts-inter-1st-year-maths-1a-matrices-solutions-exercise-3f","status":"publish","type":"post","link":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-matrices-solutions-exercise-3f\/","title":{"rendered":"TS Inter 1st Year Maths 1A Matrices Solutions Exercise 3(f)"},"content":{"rendered":"\n<h3>I.<br> Find the rank of each of the following matrices.<\/h3>\n\n<h3>Question 1.<br> <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: 4.01em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.451em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.773em, 1003.22em, 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: 2.006em; 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, 1000.42em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><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;\">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: -3.26em; left: 0em;\"><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;\">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, 1000.47em, 4.942em, -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, 1000.47em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><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;\">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: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-15\"><span class=\"mrow\" id=\"MathJax-Span-16\"><span class=\"mn\" id=\"MathJax-Span-17\" 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><\/span><span class=\"mo\" id=\"MathJax-Span-18\" 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}{ll}  1 & 0 \\\\  0 & 0  \\end{array}\\right]<\/script><\/h3>\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-19\" style=\"width: 4.01em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.451em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.773em, 1003.22em, 4.429em, -999.998em); top: -3.353em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-20\"><span class=\"mrow\" id=\"MathJax-Span-21\"><span class=\"mo\" id=\"MathJax-Span-22\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-23\"><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, 4.942em, -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, 1000.42em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-24\"><span class=\"mrow\" id=\"MathJax-Span-25\"><span class=\"mn\" id=\"MathJax-Span-26\" 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: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-30\"><span class=\"mrow\" id=\"MathJax-Span-31\"><span class=\"mn\" id=\"MathJax-Span-32\" 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, 1000.47em, 4.942em, -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, 1000.47em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-27\"><span class=\"mrow\" id=\"MathJax-Span-28\"><span class=\"mn\" id=\"MathJax-Span-29\" 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: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-33\"><span class=\"mrow\" id=\"MathJax-Span-34\"><span class=\"mn\" id=\"MathJax-Span-35\" 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><\/span><span class=\"mo\" id=\"MathJax-Span-36\" 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}{ll}  1 & 0 \\\\  0 & 0  \\end{array}\\right]<\/script> and det A = 1<br> \u2234 \u03c1(A) = Rank of the matrix A = 1.<\/p>\n\n<h3>Question 2.<br> <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-3-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-37\" style=\"width: 4.01em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.451em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.773em, 1003.22em, 4.429em, -999.998em); top: -3.353em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-38\"><span class=\"mrow\" id=\"MathJax-Span-39\"><span class=\"mo\" id=\"MathJax-Span-40\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-41\"><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, 4.942em, -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, 1000.42em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-42\"><span class=\"mrow\" id=\"MathJax-Span-43\"><span class=\"mn\" id=\"MathJax-Span-44\" 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: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-48\"><span class=\"mrow\" id=\"MathJax-Span-49\"><span class=\"mn\" id=\"MathJax-Span-50\" 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, 1000.47em, 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, 1000.47em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-45\"><span class=\"mrow\" id=\"MathJax-Span-46\"><span class=\"mn\" id=\"MathJax-Span-47\" 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: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-51\"><span class=\"mrow\" id=\"MathJax-Span-52\"><span class=\"mn\" id=\"MathJax-Span-53\" 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-54\" 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}{ll}  1 & 0 \\\\  0 & 1  \\end{array}\\right]<\/script><\/h3>\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-55\" style=\"width: 4.01em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.451em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.773em, 1003.22em, 4.429em, -999.998em); top: -3.353em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-56\"><span class=\"mrow\" id=\"MathJax-Span-57\"><span class=\"mo\" id=\"MathJax-Span-58\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-59\"><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, 4.942em, -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, 1000.42em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-60\"><span class=\"mrow\" id=\"MathJax-Span-61\"><span class=\"mn\" id=\"MathJax-Span-62\" 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: -3.26em; left: 0em;\"><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;\">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, 1000.47em, 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, 1000.47em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><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;\">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: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-69\"><span class=\"mrow\" id=\"MathJax-Span-70\"><span class=\"mn\" id=\"MathJax-Span-71\" 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-72\" 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}{ll}  1 & 0 \\\\  0 & 1  \\end{array}\\right]<\/script> and det A = 1 \u2260 0.<br> \u2234 \u03c1(A) = 2<\/p>\n\n<h3>Question 3.<br> <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-5-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-73\" style=\"width: 4.01em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.451em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.773em, 1003.22em, 4.429em, -999.998em); top: -3.353em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-74\"><span class=\"mrow\" id=\"MathJax-Span-75\"><span class=\"mo\" id=\"MathJax-Span-76\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-77\"><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, 4.942em, -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, 1000.42em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-78\"><span class=\"mrow\" id=\"MathJax-Span-79\"><span class=\"mn\" id=\"MathJax-Span-80\" 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: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-84\"><span class=\"mrow\" id=\"MathJax-Span-85\"><span class=\"mn\" id=\"MathJax-Span-86\" 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, 1000.47em, 4.942em, -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-81\"><span class=\"mrow\" id=\"MathJax-Span-82\"><span class=\"mn\" id=\"MathJax-Span-83\" 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: -3.26em; left: 0em;\"><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><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-90\" 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}{ll}  1 & 1 \\\\  0 & 0  \\end{array}\\right]<\/script><\/h3>\n\n<p>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-91\" style=\"width: 4.01em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.451em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.773em, 1003.22em, 4.429em, -999.998em); top: -3.353em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-92\"><span class=\"mrow\" id=\"MathJax-Span-93\"><span class=\"mo\" id=\"MathJax-Span-94\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-95\"><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, 4.942em, -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, 1000.42em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-96\"><span class=\"mrow\" id=\"MathJax-Span-97\"><span class=\"mn\" id=\"MathJax-Span-98\" 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: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-102\"><span class=\"mrow\" id=\"MathJax-Span-103\"><span class=\"mn\" id=\"MathJax-Span-104\" 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, 1000.47em, 4.942em, -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-99\"><span class=\"mrow\" id=\"MathJax-Span-100\"><span class=\"mn\" id=\"MathJax-Span-101\" 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: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-105\"><span class=\"mrow\" id=\"MathJax-Span-106\"><span class=\"mn\" id=\"MathJax-Span-107\" 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><\/span><span class=\"mo\" id=\"MathJax-Span-108\" 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}{ll}  1 & 1 \\\\  0 & 0  \\end{array}\\right]<\/script> and det A = 0<br> \u2234 \u03c1(A) = 1<\/p>\n\n<h3>Question 4.<br> <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-7-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-109\" style=\"width: 4.01em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.451em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.773em, 1003.22em, 4.429em, -999.998em); top: -3.353em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-110\"><span class=\"mrow\" id=\"MathJax-Span-111\"><span class=\"mo\" id=\"MathJax-Span-112\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-113\"><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, 1000.42em, 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, 1000.42em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-114\"><span class=\"mrow\" id=\"MathJax-Span-115\"><span class=\"mn\" id=\"MathJax-Span-116\" 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-120\"><span class=\"mrow\" id=\"MathJax-Span-121\"><span class=\"mn\" id=\"MathJax-Span-122\" 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, 1000.47em, 4.942em, -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-117\"><span class=\"mrow\" id=\"MathJax-Span-118\"><span class=\"mn\" id=\"MathJax-Span-119\" 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: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-123\"><span class=\"mrow\" id=\"MathJax-Span-124\"><span class=\"mn\" id=\"MathJax-Span-125\" 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><\/span><span class=\"mo\" id=\"MathJax-Span-126\" 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-7\">\\left[\\begin{array}{ll}  1 & 1 \\\\  1 & 0  \\end{array}\\right]<\/script><\/h3>\n\n<p>Answer:<br> Let A = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-8-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-127\" style=\"width: 4.01em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.451em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.773em, 1003.22em, 4.429em, -999.998em); top: -3.353em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-128\"><span class=\"mrow\" id=\"MathJax-Span-129\"><span class=\"mo\" id=\"MathJax-Span-130\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-131\"><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, 1000.42em, 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, 1000.42em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-132\"><span class=\"mrow\" id=\"MathJax-Span-133\"><span class=\"mn\" id=\"MathJax-Span-134\" 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-138\"><span class=\"mrow\" id=\"MathJax-Span-139\"><span class=\"mn\" id=\"MathJax-Span-140\" 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, 1000.47em, 4.942em, -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-135\"><span class=\"mrow\" id=\"MathJax-Span-136\"><span class=\"mn\" id=\"MathJax-Span-137\" 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: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-141\"><span class=\"mrow\" id=\"MathJax-Span-142\"><span class=\"mn\" id=\"MathJax-Span-143\" 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><\/span><span class=\"mo\" id=\"MathJax-Span-144\" 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-8\">\\left[\\begin{array}{ll}  1 & 1 \\\\  1 & 0  \\end{array}\\right]<\/script> and det A = -1 \u2260 0.<\/p>\n\n<h3>Question 5.<br> <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-9-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-145\" style=\"width: 7.598em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 6.526em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.773em, 1006.29em, 4.429em, -999.998em); top: -3.353em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-146\"><span class=\"mrow\" id=\"MathJax-Span-147\"><span class=\"mo\" id=\"MathJax-Span-148\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-149\"><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(2.565em, 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, 1000.42em, 4.15em, -999.998em); top: -4.658em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-150\"><span class=\"mrow\" id=\"MathJax-Span-151\"><span class=\"mn\" id=\"MathJax-Span-152\" 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: -3.26em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-160\"><span class=\"mrow\" id=\"MathJax-Span-161\"><span class=\"mn\" id=\"MathJax-Span-162\" 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(3.218em, 1000.47em, 4.15em, -999.998em); top: -4.658em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-153\"><span class=\"mrow\" id=\"MathJax-Span-154\"><span class=\"mn\" id=\"MathJax-Span-155\" 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, 1001.21em, 4.243em, -999.998em); top: -3.26em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-163\"><span class=\"mrow\" id=\"MathJax-Span-164\"><span class=\"mo\" id=\"MathJax-Span-165\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-166\" 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, 4.942em, -999.998em); top: -4.005em; left: 3.777em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.26em, 4.243em, -999.998em); top: -4.658em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-156\"><span class=\"mrow\" id=\"MathJax-Span-157\"><span class=\"mo\" id=\"MathJax-Span-158\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-159\" 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-167\"><span class=\"mrow\" id=\"MathJax-Span-168\"><span class=\"mn\" id=\"MathJax-Span-169\" 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><\/span><span class=\"mo\" id=\"MathJax-Span-170\" 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-9\">\\left[\\begin{array}{rrr}  1 & 0 & -4 \\\\  2 & -1 & 3  \\end{array}\\right]<\/script><\/h3>\n\n<p>Answer:<br> Let A = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-10-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-171\" style=\"width: 7.598em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 6.526em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.773em, 1006.29em, 4.429em, -999.998em); top: -3.353em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-172\"><span class=\"mrow\" id=\"MathJax-Span-173\"><span class=\"mo\" id=\"MathJax-Span-174\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-175\"><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(2.565em, 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, 1000.42em, 4.15em, -999.998em); top: -4.658em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-176\"><span class=\"mrow\" id=\"MathJax-Span-177\"><span class=\"mn\" id=\"MathJax-Span-178\" 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: -3.26em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-186\"><span class=\"mrow\" id=\"MathJax-Span-187\"><span class=\"mn\" id=\"MathJax-Span-188\" 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(3.218em, 1000.47em, 4.15em, -999.998em); top: -4.658em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-179\"><span class=\"mrow\" id=\"MathJax-Span-180\"><span class=\"mn\" id=\"MathJax-Span-181\" 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, 1001.21em, 4.243em, -999.998em); top: -3.26em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-189\"><span class=\"mrow\" id=\"MathJax-Span-190\"><span class=\"mo\" id=\"MathJax-Span-191\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-192\" 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, 4.942em, -999.998em); top: -4.005em; left: 3.777em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.26em, 4.243em, -999.998em); top: -4.658em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-182\"><span class=\"mrow\" id=\"MathJax-Span-183\"><span class=\"mo\" id=\"MathJax-Span-184\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-185\" 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-193\"><span class=\"mrow\" id=\"MathJax-Span-194\"><span class=\"mn\" id=\"MathJax-Span-195\" 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><\/span><span class=\"mo\" id=\"MathJax-Span-196\" 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-10\">\\left[\\begin{array}{rrr}  1 & 0 & -4 \\\\  2 & -1 & 3  \\end{array}\\right]<\/script><br> The determinant of a submatrix of order 2 \u00d7 2 of A = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-11-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-197\" 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-198\"><span class=\"mrow\" id=\"MathJax-Span-199\"><span class=\"mo\" id=\"MathJax-Span-200\" 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-201\"><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, 1000.42em, 4.15em, -999.998em); top: -4.658em; left: 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;\">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: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-209\"><span class=\"mrow\" id=\"MathJax-Span-210\"><span class=\"mn\" id=\"MathJax-Span-211\" 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, 1001.26em, 4.942em, -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, 1001.26em, 4.243em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-205\"><span class=\"mrow\" id=\"MathJax-Span-206\"><span class=\"mo\" id=\"MathJax-Span-207\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-208\" 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-212\"><span class=\"mrow\" id=\"MathJax-Span-213\"><span class=\"mn\" id=\"MathJax-Span-214\" 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><\/span><span class=\"mo\" id=\"MathJax-Span-215\" 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}  1 & -4 \\\\  2 & 3  \\end{array}\\right|<\/script> = 3 + 8 = 11 \u2260 0<br> \u2234 \u03c1(A) = 2<\/p>\n\n<h3>Question 6.<br> <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-12-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-216\" style=\"width: 5.734em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.942em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.773em, 1004.71em, 4.429em, -999.998em); top: -3.353em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-217\"><span class=\"mrow\" id=\"MathJax-Span-218\"><span class=\"mo\" id=\"MathJax-Span-219\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-220\"><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(2.565em, 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, 1000.42em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-221\"><span class=\"mrow\" id=\"MathJax-Span-222\"><span class=\"mn\" id=\"MathJax-Span-223\" 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: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-230\"><span class=\"mrow\" id=\"MathJax-Span-231\"><span class=\"mn\" id=\"MathJax-Span-232\" 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, 1000.47em, 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, 1000.47em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-224\"><span class=\"mrow\" id=\"MathJax-Span-225\"><span class=\"mn\" id=\"MathJax-Span-226\" 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: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-233\"><span class=\"mrow\" id=\"MathJax-Span-234\"><span class=\"mn\" id=\"MathJax-Span-235\" 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, 1000.47em, 4.942em, -999.998em); top: -4.005em; left: 2.985em;\"><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: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-227\"><span class=\"mrow\" id=\"MathJax-Span-228\"><span class=\"mn\" id=\"MathJax-Span-229\" style=\"font-family: MathJax_Main;\">6<\/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-236\"><span class=\"mrow\" id=\"MathJax-Span-237\"><span class=\"mn\" id=\"MathJax-Span-238\" 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><\/span><span class=\"mo\" id=\"MathJax-Span-239\" 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-12\">\\left[\\begin{array}{lll}  1 & 3 & 6 \\\\  2 & 4 & 3  \\end{array}\\right]<\/script><\/h3>\n\n<p>Answer:<br> Let A = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-13-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-240\" style=\"width: 5.734em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.942em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.773em, 1004.71em, 4.429em, -999.998em); top: -3.353em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-241\"><span class=\"mrow\" id=\"MathJax-Span-242\"><span class=\"mo\" id=\"MathJax-Span-243\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-244\"><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(2.565em, 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, 1000.42em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-245\"><span class=\"mrow\" id=\"MathJax-Span-246\"><span class=\"mn\" id=\"MathJax-Span-247\" 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: -3.26em; 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;\">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, 1000.47em, 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, 1000.47em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-248\"><span class=\"mrow\" id=\"MathJax-Span-249\"><span class=\"mn\" id=\"MathJax-Span-250\" 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: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-257\"><span class=\"mrow\" id=\"MathJax-Span-258\"><span class=\"mn\" id=\"MathJax-Span-259\" 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, 1000.47em, 4.942em, -999.998em); top: -4.005em; left: 2.985em;\"><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: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-251\"><span class=\"mrow\" id=\"MathJax-Span-252\"><span class=\"mn\" id=\"MathJax-Span-253\" style=\"font-family: MathJax_Main;\">6<\/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-260\"><span class=\"mrow\" id=\"MathJax-Span-261\"><span class=\"mn\" id=\"MathJax-Span-262\" 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><\/span><span class=\"mo\" id=\"MathJax-Span-263\" 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-13\">\\left[\\begin{array}{lll}  1 & 3 & 6 \\\\  2 & 4 & 3  \\end{array}\\right]<\/script><br> The determinant of a submatrix order 2 \u00d7 2 of A is = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-14-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-264\" 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-265\"><span class=\"mrow\" id=\"MathJax-Span-266\"><span class=\"mo\" id=\"MathJax-Span-267\" 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-268\"><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(3.218em, 1000.42em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-269\"><span class=\"mrow\" id=\"MathJax-Span-270\"><span class=\"mn\" id=\"MathJax-Span-271\" 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: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-275\"><span class=\"mrow\" id=\"MathJax-Span-276\"><span class=\"mn\" id=\"MathJax-Span-277\" 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, 1000.47em, 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, 1000.47em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-272\"><span class=\"mrow\" id=\"MathJax-Span-273\"><span class=\"mn\" id=\"MathJax-Span-274\" 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: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-278\"><span class=\"mrow\" id=\"MathJax-Span-279\"><span class=\"mn\" id=\"MathJax-Span-280\" 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-281\" 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-14\">\\left|\\begin{array}{ll}  1 & 3 \\\\  2 & 4  \\end{array}\\right|<\/script> = -2 \u2260 0<\/p>\n\n<h3>II.<br>\n\nQuestion 1.<br> <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-15-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-282\" 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-283\"><span class=\"mrow\" id=\"MathJax-Span-284\"><span class=\"mo\" id=\"MathJax-Span-285\" 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-286\"><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, 6.107em, -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: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-287\"><span class=\"mrow\" id=\"MathJax-Span-288\"><span class=\"mn\" id=\"MathJax-Span-289\" 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-296\"><span class=\"mrow\" id=\"MathJax-Span-297\"><span class=\"mn\" id=\"MathJax-Span-298\" 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: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-305\"><span class=\"mrow\" id=\"MathJax-Span-306\"><span class=\"mn\" id=\"MathJax-Span-307\" 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 style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.107em, -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, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 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;\">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-299\"><span class=\"mrow\" id=\"MathJax-Span-300\"><span class=\"mn\" id=\"MathJax-Span-301\" 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-308\"><span class=\"mrow\" id=\"MathJax-Span-309\"><span class=\"mn\" id=\"MathJax-Span-310\" 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 style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.06em, -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, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-293\"><span class=\"mrow\" id=\"MathJax-Span-294\"><span class=\"mn\" id=\"MathJax-Span-295\" 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; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-302\"><span class=\"mrow\" id=\"MathJax-Span-303\"><span class=\"mn\" id=\"MathJax-Span-304\" 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-311\"><span class=\"mrow\" id=\"MathJax-Span-312\"><span class=\"mn\" id=\"MathJax-Span-313\" 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-314\" 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-15\">\\left[\\begin{array}{lll}  1 & 0 & 0 \\\\  0 & 1 & 0 \\\\  0 & 0 & 1  \\end{array}\\right]<\/script><\/h3>\n\n<p>Answer:<br> Let A = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-16-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-315\" 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-316\"><span class=\"mrow\" id=\"MathJax-Span-317\"><span class=\"mo\" id=\"MathJax-Span-318\" 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-319\"><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, 6.107em, -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: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-320\"><span class=\"mrow\" id=\"MathJax-Span-321\"><span class=\"mn\" id=\"MathJax-Span-322\" 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-329\"><span class=\"mrow\" id=\"MathJax-Span-330\"><span class=\"mn\" id=\"MathJax-Span-331\" 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: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-338\"><span class=\"mrow\" id=\"MathJax-Span-339\"><span class=\"mn\" id=\"MathJax-Span-340\" 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 style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.107em, -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, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-323\"><span class=\"mrow\" id=\"MathJax-Span-324\"><span class=\"mn\" id=\"MathJax-Span-325\" 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-332\"><span class=\"mrow\" id=\"MathJax-Span-333\"><span class=\"mn\" id=\"MathJax-Span-334\" 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-341\"><span class=\"mrow\" id=\"MathJax-Span-342\"><span class=\"mn\" id=\"MathJax-Span-343\" 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 style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.06em, -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, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-326\"><span class=\"mrow\" id=\"MathJax-Span-327\"><span class=\"mn\" id=\"MathJax-Span-328\" 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; left: 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;\">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-344\"><span class=\"mrow\" id=\"MathJax-Span-345\"><span class=\"mn\" id=\"MathJax-Span-346\" 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-347\" 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-16\">\\left[\\begin{array}{lll}  1 & 0 & 0 \\\\  0 & 1 & 0 \\\\  0 & 0 & 1  \\end{array}\\right]<\/script> and det A = 1(1 \u2013 0) = 1 \u2260 0<br> \u2234 \u03c1(A) = 3<\/p>\n\n<h3>Question 2.<br> <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-17-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-348\" 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-349\"><span class=\"mrow\" id=\"MathJax-Span-350\"><span class=\"mo\" id=\"MathJax-Span-351\" 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-352\"><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, 6.107em, -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-353\"><span class=\"mrow\" id=\"MathJax-Span-354\"><span class=\"mn\" id=\"MathJax-Span-355\" 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-363\"><span class=\"mrow\" id=\"MathJax-Span-364\"><span class=\"mn\" id=\"MathJax-Span-365\" 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; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-372\"><span class=\"mrow\" id=\"MathJax-Span-373\"><span class=\"mn\" id=\"MathJax-Span-374\" 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 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(3.171em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-356\"><span class=\"mrow\" id=\"MathJax-Span-357\"><span class=\"mn\" id=\"MathJax-Span-358\" 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-366\"><span class=\"mrow\" id=\"MathJax-Span-367\"><span class=\"mn\" id=\"MathJax-Span-368\" 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, 1000.42em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-375\"><span class=\"mrow\" id=\"MathJax-Span-376\"><span class=\"mn\" id=\"MathJax-Span-377\" 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, 1001.21em, 6.06em, -999.998em); top: -4.471em; left: 2.985em;\"><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; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-359\"><span class=\"mrow\" id=\"MathJax-Span-360\"><span class=\"mo\" id=\"MathJax-Span-361\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-362\" 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-369\"><span class=\"mrow\" id=\"MathJax-Span-370\"><span class=\"mn\" id=\"MathJax-Span-371\" 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-378\"><span class=\"mrow\" id=\"MathJax-Span-379\"><span class=\"mn\" id=\"MathJax-Span-380\" 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><\/span><span class=\"mo\" id=\"MathJax-Span-381\" 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-17\">\\left[\\begin{array}{ccc}  1 & 4 & -1 \\\\  2 & 3 & 0 \\\\  0 & 1 & 2  \\end{array}\\right]<\/script><\/h3>\n\n<p>Answer:<br> Let A = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-18-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-382\" 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-383\"><span class=\"mrow\" id=\"MathJax-Span-384\"><span class=\"mo\" id=\"MathJax-Span-385\" 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-386\"><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, 6.107em, -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-387\"><span class=\"mrow\" id=\"MathJax-Span-388\"><span class=\"mn\" id=\"MathJax-Span-389\" 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-397\"><span class=\"mrow\" id=\"MathJax-Span-398\"><span class=\"mn\" id=\"MathJax-Span-399\" 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; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-406\"><span class=\"mrow\" id=\"MathJax-Span-407\"><span class=\"mn\" id=\"MathJax-Span-408\" 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 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(3.171em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-390\"><span class=\"mrow\" id=\"MathJax-Span-391\"><span class=\"mn\" id=\"MathJax-Span-392\" 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-400\"><span class=\"mrow\" id=\"MathJax-Span-401\"><span class=\"mn\" id=\"MathJax-Span-402\" 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, 1000.42em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-409\"><span class=\"mrow\" id=\"MathJax-Span-410\"><span class=\"mn\" id=\"MathJax-Span-411\" 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, 1001.21em, 6.06em, -999.998em); top: -4.471em; left: 2.985em;\"><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; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-393\"><span class=\"mrow\" id=\"MathJax-Span-394\"><span class=\"mo\" id=\"MathJax-Span-395\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-396\" 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-403\"><span class=\"mrow\" id=\"MathJax-Span-404\"><span class=\"mn\" id=\"MathJax-Span-405\" 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-412\"><span class=\"mrow\" id=\"MathJax-Span-413\"><span class=\"mn\" id=\"MathJax-Span-414\" 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><\/span><span class=\"mo\" id=\"MathJax-Span-415\" 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}{ccc}  1 & 4 & -1 \\\\  2 & 3 & 0 \\\\  0 & 1 & 2  \\end{array}\\right]<\/script><br> and det A = 1(6) \u2013 4(4) \u2013 1(2)<br> = 6 \u2013 16 \u2013 12 = -12 \u2260 0<br> \u2234 \u03c1(A) = 3<\/p>\n\n<h3>Question 3.<br> <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-19-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-416\" 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-417\"><span class=\"mrow\" id=\"MathJax-Span-418\"><span class=\"mo\" id=\"MathJax-Span-419\" 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-420\"><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, 6.107em, -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: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-421\"><span class=\"mrow\" id=\"MathJax-Span-422\"><span class=\"mn\" id=\"MathJax-Span-423\" 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-430\"><span class=\"mrow\" id=\"MathJax-Span-431\"><span class=\"mn\" id=\"MathJax-Span-432\" 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; left: 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;\">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 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, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-424\"><span class=\"mrow\" id=\"MathJax-Span-425\"><span class=\"mn\" id=\"MathJax-Span-426\" 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: -3.959em; left: 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;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-442\"><span class=\"mrow\" id=\"MathJax-Span-443\"><span class=\"mn\" id=\"MathJax-Span-444\" 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: 2.985em;\"><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-427\"><span class=\"mrow\" id=\"MathJax-Span-428\"><span class=\"mn\" id=\"MathJax-Span-429\" 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; left: 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;\">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: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-445\"><span class=\"mrow\" id=\"MathJax-Span-446\"><span class=\"mn\" id=\"MathJax-Span-447\" 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><\/span><span class=\"mo\" id=\"MathJax-Span-448\" 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}{lll}  1 & 2 & 3 \\\\  2 & 3 & 4 \\\\  0 & 1 & 2  \\end{array}\\right]<\/script> (March 2015 T.S)<\/h3>\n\n<p> Answer:<br> Let A = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-20-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-449\" 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-450\"><span class=\"mrow\" id=\"MathJax-Span-451\"><span class=\"mo\" id=\"MathJax-Span-452\" 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-453\"><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, 6.107em, -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: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-454\"><span class=\"mrow\" id=\"MathJax-Span-455\"><span class=\"mn\" id=\"MathJax-Span-456\" 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-463\"><span class=\"mrow\" id=\"MathJax-Span-464\"><span class=\"mn\" id=\"MathJax-Span-465\" 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; left: 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;\">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 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, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-457\"><span class=\"mrow\" id=\"MathJax-Span-458\"><span class=\"mn\" id=\"MathJax-Span-459\" 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: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-466\"><span class=\"mrow\" id=\"MathJax-Span-467\"><span class=\"mn\" id=\"MathJax-Span-468\" 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, 1000.42em, 4.15em, -999.998em); top: -2.561em; left: 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;\">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: 2.985em;\"><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-460\"><span class=\"mrow\" id=\"MathJax-Span-461\"><span class=\"mn\" id=\"MathJax-Span-462\" 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; left: 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;\">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: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-478\"><span class=\"mrow\" id=\"MathJax-Span-479\"><span class=\"mn\" id=\"MathJax-Span-480\" 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><\/span><span class=\"mo\" id=\"MathJax-Span-481\" 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-20\">\\left[\\begin{array}{lll}  1 & 2 & 3 \\\\  2 & 3 & 4 \\\\  0 & 1 & 2  \\end{array}\\right]<\/script><br> and det A = 1(6 \u2013 4) \u2013 2(4) + 3(2)<br> = 2 \u2013 8 + 6 = 0<br> The determinant of submatrix of order 2 \u00d7 2 of A = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-21-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-482\" 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-483\"><span class=\"mrow\" id=\"MathJax-Span-484\"><span class=\"mo\" id=\"MathJax-Span-485\" 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-486\"><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, 4.942em, -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, 1000.47em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-487\"><span class=\"mrow\" id=\"MathJax-Span-488\"><span class=\"mn\" id=\"MathJax-Span-489\" 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: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-493\"><span class=\"mrow\" id=\"MathJax-Span-494\"><span class=\"mn\" id=\"MathJax-Span-495\" 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, 4.895em, -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, 1000.47em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-490\"><span class=\"mrow\" id=\"MathJax-Span-491\"><span class=\"mn\" id=\"MathJax-Span-492\" 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: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-496\"><span class=\"mrow\" id=\"MathJax-Span-497\"><span class=\"mn\" id=\"MathJax-Span-498\" 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-499\" 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}{ll}  2 & 3 \\\\  3 & 4  \\end{array}\\right|<\/script> = 8 \u2013 9 = \u2013 1 \u2260 0<br> Hence \u03c1(A) = 2<\/p>\n\n<h3>Question 4.<br> <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-22-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-500\" 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-501\"><span class=\"mrow\" id=\"MathJax-Span-502\"><span class=\"mo\" id=\"MathJax-Span-503\" 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-504\"><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, 1000.42em, 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-505\"><span class=\"mrow\" id=\"MathJax-Span-506\"><span class=\"mn\" id=\"MathJax-Span-507\" 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-514\"><span class=\"mrow\" id=\"MathJax-Span-515\"><span class=\"mn\" id=\"MathJax-Span-516\" 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-523\"><span class=\"mrow\" id=\"MathJax-Span-524\"><span class=\"mn\" id=\"MathJax-Span-525\" 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, 1000.42em, 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-508\"><span class=\"mrow\" id=\"MathJax-Span-509\"><span class=\"mn\" id=\"MathJax-Span-510\" 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-517\"><span class=\"mrow\" id=\"MathJax-Span-518\"><span class=\"mn\" id=\"MathJax-Span-519\" 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-526\"><span class=\"mrow\" id=\"MathJax-Span-527\"><span class=\"mn\" id=\"MathJax-Span-528\" 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, 1000.42em, 6.06em, -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-511\"><span class=\"mrow\" id=\"MathJax-Span-512\"><span class=\"mn\" id=\"MathJax-Span-513\" 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-520\"><span class=\"mrow\" id=\"MathJax-Span-521\"><span class=\"mn\" id=\"MathJax-Span-522\" 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-529\"><span class=\"mrow\" id=\"MathJax-Span-530\"><span class=\"mn\" id=\"MathJax-Span-531\" 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-532\" 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-22\">\\left[\\begin{array}{lll}  1 & 1 & 1 \\\\  1 & 1 & 1 \\\\  1 & 1 & 1  \\end{array}\\right]<\/script><\/h3>\n\n<p>Answer:<br> Let A = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-23-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-533\" 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-534\"><span class=\"mrow\" id=\"MathJax-Span-535\"><span class=\"mo\" id=\"MathJax-Span-536\" 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-537\"><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, 1000.42em, 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-538\"><span class=\"mrow\" id=\"MathJax-Span-539\"><span class=\"mn\" id=\"MathJax-Span-540\" 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-547\"><span class=\"mrow\" id=\"MathJax-Span-548\"><span class=\"mn\" id=\"MathJax-Span-549\" 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-556\"><span class=\"mrow\" id=\"MathJax-Span-557\"><span class=\"mn\" id=\"MathJax-Span-558\" 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, 1000.42em, 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-541\"><span class=\"mrow\" id=\"MathJax-Span-542\"><span class=\"mn\" id=\"MathJax-Span-543\" 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-550\"><span class=\"mrow\" id=\"MathJax-Span-551\"><span class=\"mn\" id=\"MathJax-Span-552\" 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-559\"><span class=\"mrow\" id=\"MathJax-Span-560\"><span class=\"mn\" id=\"MathJax-Span-561\" 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, 1000.42em, 6.06em, -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-544\"><span class=\"mrow\" id=\"MathJax-Span-545\"><span class=\"mn\" id=\"MathJax-Span-546\" 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-553\"><span class=\"mrow\" id=\"MathJax-Span-554\"><span class=\"mn\" id=\"MathJax-Span-555\" 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-562\"><span class=\"mrow\" id=\"MathJax-Span-563\"><span class=\"mn\" id=\"MathJax-Span-564\" 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-565\" 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-23\">\\left[\\begin{array}{lll}  1 & 1 & 1 \\\\  1 & 1 & 1 \\\\  1 & 1 & 1  \\end{array}\\right]<\/script> and<br> det A = 1(0) \u2013 1(0) + 1(0) = 0<br> The determinant of submatrix of order 2 \u00d7 2 of A is <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-24-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-566\" 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-567\"><span class=\"mrow\" id=\"MathJax-Span-568\"><span class=\"mo\" id=\"MathJax-Span-569\" 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-570\"><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, 1000.42em, 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, 1000.42em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-571\"><span class=\"mrow\" id=\"MathJax-Span-572\"><span class=\"mn\" id=\"MathJax-Span-573\" 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-577\"><span class=\"mrow\" id=\"MathJax-Span-578\"><span class=\"mn\" id=\"MathJax-Span-579\" 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-574\"><span class=\"mrow\" id=\"MathJax-Span-575\"><span class=\"mn\" id=\"MathJax-Span-576\" 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-580\"><span class=\"mrow\" id=\"MathJax-Span-581\"><span class=\"mn\" id=\"MathJax-Span-582\" 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-583\" 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-24\">\\left|\\begin{array}{ll}  1 & 1 \\\\  1 & 1  \\end{array}\\right|<\/script> = 0<br> Hence \u03c1(A) = 1<\/p>\n\n<h3>Question 5.<br> <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-25-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-584\" style=\"width: 9.602em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 8.25em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1007.92em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-585\"><span class=\"mrow\" id=\"MathJax-Span-586\"><span class=\"mo\" id=\"MathJax-Span-587\" 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-588\"><span style=\"display: inline-block; position: relative; width: 6.573em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.425em, 1001.21em, 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-589\"><span class=\"mrow\" id=\"MathJax-Span-590\"><span class=\"mn\" id=\"MathJax-Span-591\" 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-602\"><span class=\"mrow\" id=\"MathJax-Span-603\"><span class=\"mn\" id=\"MathJax-Span-604\" 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-614\"><span class=\"mrow\" id=\"MathJax-Span-615\"><span class=\"mo\" id=\"MathJax-Span-616\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-617\" 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: 4.569em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.107em, -999.998em); top: -4.471em; 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: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-592\"><span class=\"mrow\" id=\"MathJax-Span-593\"><span class=\"mn\" id=\"MathJax-Span-594\" 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-605\"><span class=\"mrow\" id=\"MathJax-Span-606\"><span class=\"mn\" id=\"MathJax-Span-607\" 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-618\"><span class=\"mrow\" id=\"MathJax-Span-619\"><span class=\"mn\" id=\"MathJax-Span-620\" 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, 1000.47em, 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(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-595\"><span class=\"mrow\" id=\"MathJax-Span-596\"><span class=\"mn\" id=\"MathJax-Span-597\" 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; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-608\"><span class=\"mrow\" id=\"MathJax-Span-609\"><span class=\"mn\" id=\"MathJax-Span-610\" 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; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-621\"><span class=\"mrow\" id=\"MathJax-Span-622\"><span class=\"mn\" id=\"MathJax-Span-623\" 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, 6.107em, -999.998em); top: -4.471em; left: 5.268em;\"><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-598\"><span class=\"mrow\" id=\"MathJax-Span-599\"><span class=\"mo\" id=\"MathJax-Span-600\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-601\" 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-611\"><span class=\"mrow\" id=\"MathJax-Span-612\"><span class=\"mn\" id=\"MathJax-Span-613\" 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-624\"><span class=\"mrow\" id=\"MathJax-Span-625\"><span class=\"mn\" id=\"MathJax-Span-626\" 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><\/span><span class=\"mo\" id=\"MathJax-Span-627\" 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-25\">\\left[\\begin{array}{rrrr}  1 & 2 & 0 & -1 \\\\  3 & 4 & 1 & 2 \\\\  -2 & 3 & 2 & 5  \\end{array}\\right]<\/script><\/h3>\n\n<p>Answer:<br> Consider 3 \u00d7 3 submatrix of above matrix<br> |A| = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-26-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-628\" style=\"width: 6.06em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 5.221em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1005.08em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-629\"><span class=\"mrow\" id=\"MathJax-Span-630\"><span class=\"mo\" id=\"MathJax-Span-631\" 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-632\"><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(2.425em, 1001.21em, 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; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-633\"><span class=\"mrow\" id=\"MathJax-Span-634\"><span class=\"mn\" id=\"MathJax-Span-635\" 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-642\"><span class=\"mrow\" id=\"MathJax-Span-643\"><span class=\"mn\" id=\"MathJax-Span-644\" 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; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-651\"><span class=\"mrow\" id=\"MathJax-Span-652\"><span class=\"mo\" id=\"MathJax-Span-653\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-654\" 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: 4.569em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.107em, -999.998em); top: -4.471em; 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: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-636\"><span class=\"mrow\" id=\"MathJax-Span-637\"><span class=\"mn\" id=\"MathJax-Span-638\" 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; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-645\"><span class=\"mrow\" id=\"MathJax-Span-646\"><span class=\"mn\" id=\"MathJax-Span-647\" 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-655\"><span class=\"mrow\" id=\"MathJax-Span-656\"><span class=\"mn\" id=\"MathJax-Span-657\" 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, 1000.47em, 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(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-639\"><span class=\"mrow\" id=\"MathJax-Span-640\"><span class=\"mn\" id=\"MathJax-Span-641\" 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: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-648\"><span class=\"mrow\" id=\"MathJax-Span-649\"><span class=\"mn\" id=\"MathJax-Span-650\" 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: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-658\"><span class=\"mrow\" id=\"MathJax-Span-659\"><span class=\"mn\" id=\"MathJax-Span-660\" 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><\/span><span class=\"mo\" id=\"MathJax-Span-661\" 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-26\">\\left|\\begin{array}{ccc}  1 & 2 & 0 \\\\  3 & 4 & 1 \\\\  -2 & 3 & 2  \\end{array}\\right|<\/script><br> = 1(8 \u2013 3) \u2013 2(9 + 8)<br> = 5 \u2013 34 = -29 \u2260 0<br> \u2234 \u03c1(A) = 3<\/p>\n\n<h3>Question 6.<br> <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-27-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-662\" style=\"width: 8.67em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 7.458em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1007.13em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-663\"><span class=\"mrow\" id=\"MathJax-Span-664\"><span class=\"mo\" id=\"MathJax-Span-665\" 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-666\"><span style=\"display: inline-block; position: relative; width: 5.781em; 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-667\"><span class=\"mrow\" id=\"MathJax-Span-668\"><span class=\"mn\" id=\"MathJax-Span-669\" 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-680\"><span class=\"mrow\" id=\"MathJax-Span-681\"><span class=\"mn\" id=\"MathJax-Span-682\" 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-692\"><span class=\"mrow\" id=\"MathJax-Span-693\"><span class=\"mn\" id=\"MathJax-Span-694\" 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: 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; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-670\"><span class=\"mrow\" id=\"MathJax-Span-671\"><span class=\"mn\" id=\"MathJax-Span-672\" 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-683\"><span class=\"mrow\" id=\"MathJax-Span-684\"><span class=\"mn\" id=\"MathJax-Span-685\" 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; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-695\"><span class=\"mrow\" id=\"MathJax-Span-696\"><span class=\"mn\" id=\"MathJax-Span-697\" 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; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-673\"><span class=\"mrow\" id=\"MathJax-Span-674\"><span class=\"mn\" id=\"MathJax-Span-675\" 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-686\"><span class=\"mrow\" id=\"MathJax-Span-687\"><span class=\"mn\" id=\"MathJax-Span-688\" 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-698\"><span class=\"mrow\" id=\"MathJax-Span-699\"><span class=\"mn\" id=\"MathJax-Span-700\" 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, 1001.21em, 6.06em, -999.998em); top: -4.471em; left: 4.522em;\"><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-676\"><span class=\"mrow\" id=\"MathJax-Span-677\"><span class=\"mo\" id=\"MathJax-Span-678\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-679\" 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: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-689\"><span class=\"mrow\" id=\"MathJax-Span-690\"><span class=\"mn\" id=\"MathJax-Span-691\" 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, 1000.42em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-701\"><span class=\"mrow\" id=\"MathJax-Span-702\"><span class=\"mn\" id=\"MathJax-Span-703\" 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-704\" 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-27\">\\left[\\begin{array}{rrrr}  0 & 1 & 1 & -2 \\\\  4 & 0 & 2 & 5 \\\\  2 & 1 & 3 & 1  \\end{array}\\right]<\/script>\n\n<\/h3>\n\n<p>Answer:<br> Let A = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-28-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-705\" style=\"width: 8.67em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 7.458em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1007.13em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-706\"><span class=\"mrow\" id=\"MathJax-Span-707\"><span class=\"mo\" id=\"MathJax-Span-708\" 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-709\"><span style=\"display: inline-block; position: relative; width: 5.781em; 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-710\"><span class=\"mrow\" id=\"MathJax-Span-711\"><span class=\"mn\" id=\"MathJax-Span-712\" 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-723\"><span class=\"mrow\" id=\"MathJax-Span-724\"><span class=\"mn\" id=\"MathJax-Span-725\" 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-735\"><span class=\"mrow\" id=\"MathJax-Span-736\"><span class=\"mn\" id=\"MathJax-Span-737\" 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: 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; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-713\"><span class=\"mrow\" id=\"MathJax-Span-714\"><span class=\"mn\" id=\"MathJax-Span-715\" 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-726\"><span class=\"mrow\" id=\"MathJax-Span-727\"><span class=\"mn\" id=\"MathJax-Span-728\" 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; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-738\"><span class=\"mrow\" id=\"MathJax-Span-739\"><span class=\"mn\" id=\"MathJax-Span-740\" 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; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-716\"><span class=\"mrow\" id=\"MathJax-Span-717\"><span class=\"mn\" id=\"MathJax-Span-718\" 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-729\"><span class=\"mrow\" id=\"MathJax-Span-730\"><span class=\"mn\" id=\"MathJax-Span-731\" 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-741\"><span class=\"mrow\" id=\"MathJax-Span-742\"><span class=\"mn\" id=\"MathJax-Span-743\" 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, 1001.21em, 6.06em, -999.998em); top: -4.471em; left: 4.522em;\"><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-719\"><span class=\"mrow\" id=\"MathJax-Span-720\"><span class=\"mo\" id=\"MathJax-Span-721\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-722\" 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: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-732\"><span class=\"mrow\" id=\"MathJax-Span-733\"><span class=\"mn\" id=\"MathJax-Span-734\" 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, 1000.42em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-744\"><span class=\"mrow\" id=\"MathJax-Span-745\"><span class=\"mn\" id=\"MathJax-Span-746\" 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-747\" 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-28\">\\left[\\begin{array}{rrrr}  0 & 1 & 1 & -2 \\\\  4 & 0 & 2 & 5 \\\\  2 & 1 & 3 & 1  \\end{array}\\right]<\/script> and<br> Consider a submatrix B of order 3 \u00d7 3 of above matrix \u2018A\u2019.<br> Then |B| = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-29-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-748\" style=\"width: 5.128em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.429em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1004.29em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-749\"><span class=\"mrow\" id=\"MathJax-Span-750\"><span class=\"mo\" id=\"MathJax-Span-751\" 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-752\"><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-753\"><span class=\"mrow\" id=\"MathJax-Span-754\"><span class=\"mn\" id=\"MathJax-Span-755\" 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; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-762\"><span class=\"mrow\" id=\"MathJax-Span-763\"><span class=\"mn\" id=\"MathJax-Span-764\" 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: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-771\"><span class=\"mrow\" id=\"MathJax-Span-772\"><span class=\"mn\" id=\"MathJax-Span-773\" 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: 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-756\"><span class=\"mrow\" id=\"MathJax-Span-757\"><span class=\"mn\" id=\"MathJax-Span-758\" 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-765\"><span class=\"mrow\" id=\"MathJax-Span-766\"><span class=\"mn\" id=\"MathJax-Span-767\" 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-774\"><span class=\"mrow\" id=\"MathJax-Span-775\"><span class=\"mn\" id=\"MathJax-Span-776\" 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-759\"><span class=\"mrow\" id=\"MathJax-Span-760\"><span class=\"mn\" id=\"MathJax-Span-761\" 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-768\"><span class=\"mrow\" id=\"MathJax-Span-769\"><span class=\"mn\" id=\"MathJax-Span-770\" 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; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-777\"><span class=\"mrow\" id=\"MathJax-Span-778\"><span class=\"mn\" id=\"MathJax-Span-779\" 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><\/span><span class=\"mo\" id=\"MathJax-Span-780\" 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-29\">\\left|\\begin{array}{lll}  0 & 1 & 1 \\\\  4 & 0 & 2 \\\\  2 & 1 & 3  \\end{array}\\right|<\/script><br> = -1(12 \u2013 4) + 1(4)<br> = -8 + 4 = -4<br> Hence \u03c1(A) = 3<\/p><\/div>\n","protected":false},"excerpt":{"rendered":"<p>I. Find the rank of each of the following matrices. Question 1. [1000] Answer: Let A = [1000] and det A = 1 \u2234 \u03c1(A) = Rank of the matrix A = 1. Question 2. [1001] Answer: Let A = [1001] and det A = 1 \u2260 0. \u2234 \u03c1(A) = 2 Question 3. [1010] [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":2805,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1544,15,38],"tags":[],"class_list":["post-2786","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-study-material-tg-inter","category-telangana","category-tg-inter"],"_links":{"self":[{"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/posts\/2786","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=2786"}],"version-history":[{"count":1,"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/posts\/2786\/revisions"}],"predecessor-version":[{"id":2787,"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/posts\/2786\/revisions\/2787"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/media\/2805"}],"wp:attachment":[{"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/media?parent=2786"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/categories?post=2786"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/tags?post=2786"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}