{"id":2689,"date":"2026-01-08T12:48:13","date_gmt":"2026-01-08T07:18:13","guid":{"rendered":"https:\/\/www.manabadi.co.in\/boards\/?p=2689"},"modified":"2026-01-08T13:55:52","modified_gmt":"2026-01-08T08:25:52","slug":"ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3c","status":"publish","type":"post","link":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3c\/","title":{"rendered":"TS Inter 1st Year Maths 1A Matrices Solutions Exercise 3(c)"},"content":{"rendered":"\n<h3>Question 1.<br> If A = <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: 6.666em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 5.734em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.773em, 1005.5em, 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: 4.289em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.565em, 1001.21em, 4.988em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -4.658em; right: 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;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -3.26em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-15\"><span class=\"mrow\" id=\"MathJax-Span-16\"><span class=\"mo\" id=\"MathJax-Span-17\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-18\" style=\"font-family: MathJax_Main;\">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, 4.895em, -999.998em); top: -4.005em; left: 2.286em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -4.658em; right: 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, 1000.42em, 4.15em, -999.998em); top: -3.26em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-19\"><span class=\"mrow\" id=\"MathJax-Span-20\"><span class=\"mn\" id=\"MathJax-Span-21\" 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: 3.777em;\"><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-12\"><span class=\"mrow\" id=\"MathJax-Span-13\"><span class=\"mn\" id=\"MathJax-Span-14\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.26em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-22\"><span class=\"mrow\" id=\"MathJax-Span-23\"><span class=\"mn\" id=\"MathJax-Span-24\" style=\"font-family: MathJax_Main;\">5<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-25\" 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}{rrr}  2 & 0 & 1 \\\\  -1 & 1 & 5  \\end{array}\\right]<\/script> and B = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-2-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-26\" 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-27\"><span class=\"mrow\" id=\"MathJax-Span-28\"><span class=\"mo\" id=\"MathJax-Span-29\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-30\"><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, 1001.21em, 4.942em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -4.658em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-31\"><span class=\"mrow\" id=\"MathJax-Span-32\"><span class=\"mo\" id=\"MathJax-Span-33\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-34\" 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-41\"><span class=\"mrow\" id=\"MathJax-Span-42\"><span class=\"mn\" id=\"MathJax-Span-43\" 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, 1000.42em, 4.895em, -999.998em); top: -4.005em; left: 2.286em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -4.658em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-35\"><span class=\"mrow\" id=\"MathJax-Span-36\"><span class=\"mn\" id=\"MathJax-Span-37\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.26em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-44\"><span class=\"mrow\" id=\"MathJax-Span-45\"><span class=\"mn\" id=\"MathJax-Span-46\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1001.21em, 4.988em, -999.998em); top: -4.005em; left: 3.777em;\"><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-38\"><span class=\"mrow\" id=\"MathJax-Span-39\"><span class=\"mn\" id=\"MathJax-Span-40\" 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-47\"><span class=\"mrow\" id=\"MathJax-Span-48\"><span class=\"mo\" id=\"MathJax-Span-49\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-50\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-51\" 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}{rrr}  -1 & 1 & 0 \\\\  0 & 1 & -2  \\end{array}\\right]<\/script> then find (AB\u2019)\u2019<\/h3>\n\n<p>Answer:<br> We have (AB)\u2019 = B\u2019A\u2019<br> and (AB\u2019)\u2019 = (B\u2019)\u2019 A\u2019 = BA\u2019 (\u2235 (B )\u2019 = B)<br> \n<img fetchpriority=\"high\" decoding=\"async\" class=\"alignnone size-full wp-image-5753\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3c-1.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(c) 1\" width=\"330\" height=\"263\" sizes=\"(max-width: 330px) 100vw, 330px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(c) 1\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(c)\"><\/p>\n<h3>Question 2.<br> If A = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-3-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-52\" style=\"width: 5.221em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.476em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1004.15em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-53\"><span class=\"mrow\" id=\"MathJax-Span-54\"><span class=\"mo\" id=\"MathJax-Span-55\" 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-56\"><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(2.425em, 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, 1001.21em, 4.243em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-57\"><span class=\"mrow\" id=\"MathJax-Span-58\"><span class=\"mo\" id=\"MathJax-Span-59\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-60\" 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-64\"><span class=\"mrow\" id=\"MathJax-Span-65\"><span class=\"mn\" id=\"MathJax-Span-66\" 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, 1001.21em, 4.243em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-70\"><span class=\"mrow\" id=\"MathJax-Span-71\"><span class=\"mo\" id=\"MathJax-Span-72\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-73\" 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: 4.569em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.06em, -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, 1000.42em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-61\"><span class=\"mrow\" id=\"MathJax-Span-62\"><span class=\"mn\" id=\"MathJax-Span-63\" 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-67\"><span class=\"mrow\" id=\"MathJax-Span-68\"><span class=\"mn\" id=\"MathJax-Span-69\" 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: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-74\"><span class=\"mrow\" id=\"MathJax-Span-75\"><span class=\"mn\" id=\"MathJax-Span-76\" style=\"font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-77\" 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-3\">\\left[\\begin{array}{rr}  -2 & 1 \\\\  5 & 0 \\\\  -1 & 4  \\end{array}\\right]<\/script> and B = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-4-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-78\" style=\"width: 6.666em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 5.734em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.773em, 1005.5em, 4.429em, -999.998em); top: -3.353em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-79\"><span class=\"mrow\" id=\"MathJax-Span-80\"><span class=\"mo\" id=\"MathJax-Span-81\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-82\"><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(2.565em, 1001.21em, 4.895em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -4.658em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-83\"><span class=\"mrow\" id=\"MathJax-Span-84\"><span class=\"mo\" id=\"MathJax-Span-85\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-86\" 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: -3.26em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-93\"><span class=\"mrow\" id=\"MathJax-Span-94\"><span class=\"mn\" id=\"MathJax-Span-95\" 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: 2.286em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -4.658em; right: 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;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.26em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-96\"><span class=\"mrow\" id=\"MathJax-Span-97\"><span class=\"mn\" id=\"MathJax-Span-98\" 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: 3.777em;\"><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-90\"><span class=\"mrow\" id=\"MathJax-Span-91\"><span class=\"mn\" id=\"MathJax-Span-92\" 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-99\"><span class=\"mrow\" id=\"MathJax-Span-100\"><span class=\"mn\" id=\"MathJax-Span-101\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-102\" 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}{rrr}  -2 & 3 & 1 \\\\  4 & 0 & 2  \\end{array}\\right]<\/script> then find 2A + B\u2019 and 3B\u2019 \u2013 A.<\/h3>\n\n<p>Answer:<br> \n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-5752\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3c-2.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(c) 2\" width=\"300\" height=\"270\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(c) 2\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(c)\"><\/p>\n\n\n<h3>Question 3.<br> If A = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-5-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-103\" style=\"width: 5.781em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.988em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.773em, 1004.75em, 4.429em, -999.998em); top: -3.353em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-104\"><span class=\"mrow\" id=\"MathJax-Span-105\"><span class=\"mo\" id=\"MathJax-Span-106\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-107\"><span style=\"display: inline-block; position: relative; width: 3.544em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.565em, 1001.21em, 4.988em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -4.658em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-108\"><span class=\"mrow\" id=\"MathJax-Span-109\"><span class=\"mn\" id=\"MathJax-Span-110\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -3.26em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-115\"><span class=\"mrow\" id=\"MathJax-Span-116\"><span class=\"mo\" id=\"MathJax-Span-117\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-118\" style=\"font-family: MathJax_Main;\">5<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1001.26em, 4.942em, -999.998em); top: -4.005em; left: 2.286em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.26em, 4.243em, -999.998em); top: -4.658em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-111\"><span class=\"mrow\" id=\"MathJax-Span-112\"><span class=\"mo\" id=\"MathJax-Span-113\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-114\" 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: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-119\"><span class=\"mrow\" id=\"MathJax-Span-120\"><span class=\"mn\" id=\"MathJax-Span-121\" 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-122\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">]<\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 3.357em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.132em; border-left: 0px solid; width: 0px; height: 2.922em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-5\">\\left[\\begin{array}{cc}  2 & -4 \\\\  -5 & 3  \\end{array}\\right]<\/script> then find A + A\u2019 and A. A\u2019 (May 2007) (Board Model Paper)<\/h3>\n\n<p>Answer:<br> \n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-5751\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3c-3.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(c) 3\" width=\"325\" height=\"237\" sizes=\"auto, (max-width: 325px) 100vw, 325px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(c) 3\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(c)\"><\/p>\n\n<h3>Question 4.<br> If 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-123\" style=\"width: 6.992em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 6.014em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1005.69em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-124\"><span class=\"mrow\" id=\"MathJax-Span-125\"><span class=\"mo\" id=\"MathJax-Span-126\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Size4; top: -2.84em; left: 0em;\">\u23a1<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Size4; top: -0.836em; left: 0em;\">\u23a3<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Size4; position: absolute; top: -1.815em; left: 0em;\">\u23a2<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-127\"><span style=\"display: inline-block; position: relative; width: 4.336em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1001.21em, 6.107em, -999.998em); top: -4.471em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-128\"><span class=\"mrow\" id=\"MathJax-Span-129\"><span class=\"mo\" id=\"MathJax-Span-130\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-131\" 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-138\"><span class=\"mrow\" id=\"MathJax-Span-139\"><span class=\"mn\" id=\"MathJax-Span-140\" 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-147\"><span class=\"mrow\" id=\"MathJax-Span-148\"><span class=\"mn\" id=\"MathJax-Span-149\" 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, 1000.51em, 6.06em, -999.998em); top: -4.471em; left: 2.286em;\"><span style=\"display: inline-block; position: relative; width: 0.562em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><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;\">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: 50%; margin-left: -0.231em;\"><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;\">5<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.277em;\"><span class=\"mtd\" id=\"MathJax-Span-150\"><span class=\"mrow\" id=\"MathJax-Span-151\"><span class=\"mi\" id=\"MathJax-Span-152\" style=\"font-family: MathJax_Math-italic;\">x<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.107em, -999.998em); top: -4.471em; left: 3.87em;\"><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-135\"><span class=\"mrow\" id=\"MathJax-Span-136\"><span class=\"mn\" id=\"MathJax-Span-137\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-144\"><span class=\"mrow\" id=\"MathJax-Span-145\"><span class=\"mn\" id=\"MathJax-Span-146\" 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(3.171em, 1000.47em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.231em;\"><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;\">7<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-156\" 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-6\">\\left[\\begin{array}{ccc}  -1 & 2 & 3 \\\\  2 & 5 & 6 \\\\  3 & x & 7  \\end{array}\\right]<\/script> is a symmetric matrix then find x.<\/h3>\n\n<p>Answer:<br> A matrix \u2018A\u2019 is said to be symmetric if A\u2019 = A<br> \n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-5750\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3c-4.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(c) 4\" width=\"292\" height=\"205\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(c) 4\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(c)\"><\/p>\n\n<h3>Question 5.<br> If A = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-7-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-157\" style=\"width: 7.924em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 6.806em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1006.48em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-158\"><span class=\"mrow\" id=\"MathJax-Span-159\"><span class=\"mo\" id=\"MathJax-Span-160\" 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-161\"><span style=\"display: inline-block; position: relative; width: 5.128em; 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, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-162\"><span class=\"mrow\" id=\"MathJax-Span-163\"><span class=\"mn\" id=\"MathJax-Span-164\" 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: -3.959em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-171\"><span class=\"mrow\" id=\"MathJax-Span-172\"><span class=\"mo\" id=\"MathJax-Span-173\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-174\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.65em;\"><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;\">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: 4.569em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.51em, 6.06em, -999.998em); top: -4.471em; left: 2.286em;\"><span style=\"display: inline-block; position: relative; width: 0.562em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-165\"><span class=\"mrow\" id=\"MathJax-Span-166\"><span class=\"mn\" id=\"MathJax-Span-167\" 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: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-175\"><span class=\"mrow\" id=\"MathJax-Span-176\"><span class=\"mn\" id=\"MathJax-Span-177\" 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(3.451em, 1000.51em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.277em;\"><span class=\"mtd\" id=\"MathJax-Span-186\"><span class=\"mrow\" id=\"MathJax-Span-187\"><span class=\"mi\" id=\"MathJax-Span-188\" style=\"font-family: MathJax_Math-italic;\">x<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1001.21em, 6.107em, -999.998em); top: -4.471em; left: 3.87em;\"><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-168\"><span class=\"mrow\" id=\"MathJax-Span-169\"><span class=\"mn\" id=\"MathJax-Span-170\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-178\"><span class=\"mrow\" id=\"MathJax-Span-179\"><span class=\"mo\" id=\"MathJax-Span-180\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-181\" 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-189\"><span class=\"mrow\" id=\"MathJax-Span-190\"><span class=\"mn\" id=\"MathJax-Span-191\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-192\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Size4; top: -2.84em; left: 0em;\">\u23a4<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Size4; top: -0.836em; left: 0em;\">\u23a6<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Size4; position: absolute; top: -1.815em; left: 0em;\">\u23a5<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-7\">\\left[\\begin{array}{ccc}  0 & 2 & 1 \\\\  -2 & 0 & -2 \\\\  -1 & x & 0  \\end{array}\\right]<\/script> is a skew symmetric matrix, find x. (May 2014, 11)<\/h3>\n\n<p>Answer:<br> A matrix A is said to be skew symmetric if A\u2019 = \u2013 A<br> <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-8-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-193\" style=\"width: 17.244em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 14.867em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.192em, 1014.54em, 6.247em, -999.998em); top: -4.471em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-194\"><span class=\"mrow\" id=\"MathJax-Span-195\"><span class=\"mo\" id=\"MathJax-Span-196\" 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-197\"><span style=\"display: inline-block; position: relative; width: 5.082em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.06em, -999.998em); top: -4.471em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-198\"><span class=\"mrow\" id=\"MathJax-Span-199\"><span class=\"mn\" id=\"MathJax-Span-200\" 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: 50%; margin-left: -0.231em;\"><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 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-220\"><span class=\"mrow\" id=\"MathJax-Span-221\"><span class=\"mn\" id=\"MathJax-Span-222\" 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(2.425em, 1001.21em, 6.247em, -999.998em); top: -4.564em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-201\"><span class=\"mrow\" id=\"MathJax-Span-202\"><span class=\"mo\" id=\"MathJax-Span-203\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-204\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.231em;\"><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;\">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: -2.561em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-223\"><span class=\"mrow\" id=\"MathJax-Span-224\"><span class=\"mo\" id=\"MathJax-Span-225\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-226\" 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, 1001.21em, 6.107em, -999.998em); top: -4.471em; left: 3.777em;\"><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-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;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.277em;\"><span class=\"mtd\" id=\"MathJax-Span-215\"><span class=\"mrow\" id=\"MathJax-Span-216\"><span class=\"texatom\" id=\"MathJax-Span-217\"><span class=\"mrow\" id=\"MathJax-Span-218\"><span class=\"mi\" id=\"MathJax-Span-219\" style=\"font-family: MathJax_Main;\">x<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.231em;\"><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;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-230\" 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 class=\"mo\" id=\"MathJax-Span-231\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"mrow\" id=\"MathJax-Span-232\" style=\"padding-left: 0.282em;\"><span class=\"mo\" id=\"MathJax-Span-233\" 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-234\"><span style=\"display: inline-block; position: relative; width: 5.082em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.06em, -999.998em); top: -4.471em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-235\"><span class=\"mrow\" id=\"MathJax-Span-236\"><span class=\"mn\" id=\"MathJax-Span-237\" style=\"font-family: MathJax_Main;\">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: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-246\"><span class=\"mrow\" id=\"MathJax-Span-247\"><span class=\"mn\" id=\"MathJax-Span-248\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-255\"><span class=\"mrow\" id=\"MathJax-Span-256\"><span class=\"mn\" id=\"MathJax-Span-257\" 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(2.425em, 1001.31em, 6.247em, -999.998em); top: -4.564em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 1.307em; 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-238\"><span class=\"mrow\" id=\"MathJax-Span-239\"><span class=\"mo\" id=\"MathJax-Span-240\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-241\" 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: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-249\"><span class=\"mrow\" id=\"MathJax-Span-250\"><span class=\"mn\" id=\"MathJax-Span-251\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.264em, 1001.31em, 4.243em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-258\"><span class=\"mrow\" id=\"MathJax-Span-259\"><span class=\"mo\" id=\"MathJax-Span-260\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"texatom\" id=\"MathJax-Span-261\"><span class=\"mrow\" id=\"MathJax-Span-262\"><span class=\"mi\" id=\"MathJax-Span-263\" style=\"font-family: MathJax_Main;\">x<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1001.21em, 6.107em, -999.998em); top: -4.471em; left: 3.823em;\"><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-242\"><span class=\"mrow\" id=\"MathJax-Span-243\"><span class=\"mo\" id=\"MathJax-Span-244\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-245\" 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-252\"><span class=\"mrow\" id=\"MathJax-Span-253\"><span class=\"mn\" id=\"MathJax-Span-254\" 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-264\"><span class=\"mrow\" id=\"MathJax-Span-265\"><span class=\"mn\" id=\"MathJax-Span-266\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-267\" 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: 4.476em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-8\">\\left[\\begin{array}{ccc}  0 & -2 & -1 \\\\  2 & 0 & \\mathrm{x} \\\\  1 & -2 & 0  \\end{array}\\right]=\\left[\\begin{array}{ccc}  0 & -2 & -1 \\\\  2 & 0 & 2 \\\\  1 & -\\mathrm{x} & 0  \\end{array}\\right]<\/script><br> from equality of matrix x = 2<\/div>\n\n<h3>Question 6.<br> Is <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-9-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-268\" style=\"width: 7.831em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 6.759em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1006.43em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-269\"><span class=\"mrow\" id=\"MathJax-Span-270\"><span class=\"mo\" id=\"MathJax-Span-271\" 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-272\"><span style=\"display: inline-block; position: relative; width: 5.082em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.425em, 1001.26em, 6.247em, -999.998em); top: -4.564em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-273\"><span class=\"mrow\" id=\"MathJax-Span-274\"><span class=\"mn\" id=\"MathJax-Span-275\" 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: -3.959em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-282\"><span class=\"mrow\" id=\"MathJax-Span-283\"><span class=\"mo\" id=\"MathJax-Span-284\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-285\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1001.26em, 4.243em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-292\"><span class=\"mrow\" id=\"MathJax-Span-293\"><span class=\"mo\" id=\"MathJax-Span-294\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-295\" 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: 4.569em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.425em, 1001.26em, 6.247em, -999.998em); top: -4.564em; left: 2.286em;\"><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-276\"><span class=\"mrow\" id=\"MathJax-Span-277\"><span class=\"mn\" id=\"MathJax-Span-278\" 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-286\"><span class=\"mrow\" id=\"MathJax-Span-287\"><span class=\"mn\" id=\"MathJax-Span-288\" 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, 1001.26em, 4.243em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-296\"><span class=\"mrow\" id=\"MathJax-Span-297\"><span class=\"mo\" id=\"MathJax-Span-298\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-299\" style=\"font-family: MathJax_Main;\">7<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.379em, 1000.47em, 6.153em, -999.998em); top: -4.518em; left: 4.569em;\"><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-279\"><span class=\"mrow\" id=\"MathJax-Span-280\"><span class=\"mn\" id=\"MathJax-Span-281\" 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(3.171em, 1000.47em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-289\"><span class=\"mrow\" id=\"MathJax-Span-290\"><span class=\"mn\" id=\"MathJax-Span-291\" style=\"font-family: MathJax_Main;\">7<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-300\"><span class=\"mrow\" id=\"MathJax-Span-301\"><span class=\"mn\" id=\"MathJax-Span-302\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.522em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-303\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Size4; top: -2.84em; left: 0em;\">\u23a4<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Size4; top: -0.836em; left: 0em;\">\u23a6<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Size4; position: absolute; top: -1.815em; left: 0em;\">\u23a5<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-9\">\\left[\\begin{array}{ccc}  0 & 1 & 4 \\\\  -1 & 0 & 7 \\\\  -4 & -7 & 0  \\end{array}\\right]<\/script> a symmetric or skew symmetric?<\/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-304\" style=\"width: 7.831em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 6.759em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1006.43em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-305\"><span class=\"mrow\" id=\"MathJax-Span-306\"><span class=\"mo\" id=\"MathJax-Span-307\" 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-308\"><span style=\"display: inline-block; position: relative; width: 5.082em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.425em, 1001.26em, 6.247em, -999.998em); top: -4.564em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-309\"><span class=\"mrow\" id=\"MathJax-Span-310\"><span class=\"mn\" id=\"MathJax-Span-311\" 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: -3.959em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-318\"><span class=\"mrow\" id=\"MathJax-Span-319\"><span class=\"mo\" id=\"MathJax-Span-320\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-321\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1001.26em, 4.243em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-328\"><span class=\"mrow\" id=\"MathJax-Span-329\"><span class=\"mo\" id=\"MathJax-Span-330\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-331\" style=\"font-family: MathJax_Main;\">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: 4.569em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.425em, 1001.26em, 6.247em, -999.998em); top: -4.564em; left: 2.286em;\"><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-312\"><span class=\"mrow\" id=\"MathJax-Span-313\"><span class=\"mn\" id=\"MathJax-Span-314\" 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-322\"><span class=\"mrow\" id=\"MathJax-Span-323\"><span class=\"mn\" id=\"MathJax-Span-324\" 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, 1001.26em, 4.243em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-332\"><span class=\"mrow\" id=\"MathJax-Span-333\"><span class=\"mo\" id=\"MathJax-Span-334\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-335\" style=\"font-family: MathJax_Main;\">7<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.379em, 1000.47em, 6.153em, -999.998em); top: -4.518em; left: 4.569em;\"><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-315\"><span class=\"mrow\" id=\"MathJax-Span-316\"><span class=\"mn\" id=\"MathJax-Span-317\" 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(3.171em, 1000.47em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-325\"><span class=\"mrow\" id=\"MathJax-Span-326\"><span class=\"mn\" id=\"MathJax-Span-327\" style=\"font-family: MathJax_Main;\">7<\/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-336\"><span class=\"mrow\" id=\"MathJax-Span-337\"><span class=\"mn\" id=\"MathJax-Span-338\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.522em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-339\" 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-10\">\\left[\\begin{array}{ccc}  0 & 1 & 4 \\\\  -1 & 0 & 7 \\\\  -4 & -7 & 0  \\end{array}\\right]<\/script> then A is symmetric if A\u2019 = A and skew symmetric if A\u2019 = \u2013 A<br> i.e., A\u2019 = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-11-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-340\" style=\"width: 17.244em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 14.867em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.192em, 1014.54em, 6.247em, -999.998em); top: -4.471em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-341\"><span class=\"mrow\" id=\"MathJax-Span-342\"><span class=\"mo\" id=\"MathJax-Span-343\" 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-344\"><span style=\"display: inline-block; position: relative; width: 5.082em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.06em, -999.998em); top: -4.471em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-345\"><span class=\"mrow\" id=\"MathJax-Span-346\"><span class=\"mn\" id=\"MathJax-Span-347\" style=\"font-family: MathJax_Main;\">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-356\"><span class=\"mrow\" id=\"MathJax-Span-357\"><span class=\"mn\" id=\"MathJax-Span-358\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -2.561em; 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;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1001.21em, 6.107em, -999.998em); top: -4.471em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-348\"><span class=\"mrow\" id=\"MathJax-Span-349\"><span class=\"mo\" id=\"MathJax-Span-350\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-351\" 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-359\"><span class=\"mrow\" id=\"MathJax-Span-360\"><span class=\"mn\" id=\"MathJax-Span-361\" 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: -2.561em; 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;\">7<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.379em, 1001.26em, 6.153em, -999.998em); top: -4.518em; 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: -5.356em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-352\"><span class=\"mrow\" id=\"MathJax-Span-353\"><span class=\"mo\" id=\"MathJax-Span-354\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-355\" 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(3.171em, 1001.26em, 4.243em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-362\"><span class=\"mrow\" id=\"MathJax-Span-363\"><span class=\"mo\" id=\"MathJax-Span-364\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-365\" style=\"font-family: MathJax_Main;\">7<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 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: 4.522em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-375\" 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 class=\"mo\" id=\"MathJax-Span-376\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"mrow\" id=\"MathJax-Span-377\" style=\"padding-left: 0.282em;\"><span class=\"mo\" id=\"MathJax-Span-378\" 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-379\"><span style=\"display: inline-block; position: relative; width: 5.082em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.425em, 1001.26em, 6.247em, -999.998em); top: -4.564em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-380\"><span class=\"mrow\" id=\"MathJax-Span-381\"><span class=\"mn\" id=\"MathJax-Span-382\" 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: -3.959em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-389\"><span class=\"mrow\" id=\"MathJax-Span-390\"><span class=\"mo\" id=\"MathJax-Span-391\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-392\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1001.26em, 4.243em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-399\"><span class=\"mrow\" id=\"MathJax-Span-400\"><span class=\"mo\" id=\"MathJax-Span-401\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-402\" 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: 4.569em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.425em, 1001.26em, 6.247em, -999.998em); top: -4.564em; left: 2.286em;\"><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-383\"><span class=\"mrow\" id=\"MathJax-Span-384\"><span class=\"mn\" id=\"MathJax-Span-385\" 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-393\"><span class=\"mrow\" id=\"MathJax-Span-394\"><span class=\"mn\" id=\"MathJax-Span-395\" 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, 1001.26em, 4.243em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-403\"><span class=\"mrow\" id=\"MathJax-Span-404\"><span class=\"mo\" id=\"MathJax-Span-405\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-406\" style=\"font-family: MathJax_Main;\">7<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.379em, 1000.47em, 6.153em, -999.998em); top: -4.518em; left: 4.569em;\"><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-386\"><span class=\"mrow\" id=\"MathJax-Span-387\"><span class=\"mn\" id=\"MathJax-Span-388\" 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(3.171em, 1000.47em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-396\"><span class=\"mrow\" id=\"MathJax-Span-397\"><span class=\"mn\" id=\"MathJax-Span-398\" style=\"font-family: MathJax_Main;\">7<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-407\"><span class=\"mrow\" id=\"MathJax-Span-408\"><span class=\"mn\" id=\"MathJax-Span-409\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.522em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-410\" 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: 4.476em;\"><\/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-11\">\\left[\\begin{array}{ccc}  0 & -1 & -4 \\\\  1 & 0 & -7 \\\\  4 & 7 & 0  \\end{array}\\right]=\\left[\\begin{array}{ccc}  0 & 1 & 4 \\\\  -1 & 0 & 7 \\\\  -4 & -7 & 0  \\end{array}\\right]<\/script> = -A<br> \u2234 The matrix A is a skew symmetric matrix.<\/p><p><im\n\n<h3>II.<br> Question 1.<br> If A = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-12-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-411\" style=\"width: 8.903em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 7.644em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.773em, 1007.41em, 4.429em, -999.998em); top: -3.353em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-412\"><span class=\"mrow\" id=\"MathJax-Span-413\"><span class=\"mo\" id=\"MathJax-Span-414\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-415\"><span style=\"display: inline-block; position: relative; width: 6.2em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.752em, 1002.99em, 4.988em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 3.031em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1002.15em, 4.15em, -999.998em); top: -4.658em; left: 50%; margin-left: -1.069em;\"><span class=\"mtd\" id=\"MathJax-Span-416\"><span class=\"mrow\" id=\"MathJax-Span-417\"><span class=\"mi\" id=\"MathJax-Span-418\" style=\"font-family: MathJax_Main;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-419\"><\/span><span class=\"mi\" id=\"MathJax-Span-420\" style=\"font-family: MathJax_Math-italic; padding-left: 0.189em;\">\u03b1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1002.99em, 4.243em, -999.998em); top: -3.26em; left: 50%; margin-left: -1.489em;\"><span class=\"mtd\" id=\"MathJax-Span-426\"><span class=\"mrow\" id=\"MathJax-Span-427\"><span class=\"mo\" id=\"MathJax-Span-428\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mi\" id=\"MathJax-Span-429\" style=\"font-family: MathJax_Main; padding-left: 0.189em;\">sin<\/span><span class=\"mo\" id=\"MathJax-Span-430\"><\/span><span class=\"mi\" id=\"MathJax-Span-431\" style=\"font-family: MathJax_Math-italic; padding-left: 0.189em;\">\u03b1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1002.15em, 4.895em, -999.998em); top: -4.005em; left: 4.01em;\"><span style=\"display: inline-block; position: relative; width: 2.192em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1002.01em, 4.15em, -999.998em); top: -4.658em; left: 50%; margin-left: -1.023em;\"><span class=\"mtd\" id=\"MathJax-Span-421\"><span class=\"mrow\" id=\"MathJax-Span-422\"><span class=\"mi\" id=\"MathJax-Span-423\" style=\"font-family: MathJax_Main;\">sin<\/span><span class=\"mo\" id=\"MathJax-Span-424\"><\/span><span class=\"mi\" id=\"MathJax-Span-425\" style=\"font-family: MathJax_Math-italic; padding-left: 0.189em;\">\u03b1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1002.15em, 4.15em, -999.998em); top: -3.26em; left: 50%; margin-left: -1.069em;\"><span class=\"mtd\" id=\"MathJax-Span-432\"><span class=\"mrow\" id=\"MathJax-Span-433\"><span class=\"mi\" id=\"MathJax-Span-434\" style=\"font-family: MathJax_Main;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-435\"><\/span><span class=\"mi\" id=\"MathJax-Span-436\" style=\"font-family: MathJax_Math-italic; padding-left: 0.189em;\">\u03b1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-437\" 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}{cc}  \\cos \\alpha & \\sin \\alpha \\\\  -\\sin \\alpha & \\cos \\alpha  \\end{array}\\right]<\/script>, show that A . A\u2019 = A\u2019 . A = I<sub>2<\/sub>. (March 2007)<\/h3>\n\n<p>Answer:<br> \n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-5748\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3c-5.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(c) 5\" width=\"378\" height=\"249\" \n sizes=\"auto, (max-width: 378px) 100vw, 378px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(c) 5\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(c)\"><\/p>\n\n<h3>Question 2.<br> If 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-438\" style=\"width: 7.831em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 6.759em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1006.43em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-439\"><span class=\"mrow\" id=\"MathJax-Span-440\"><span class=\"mo\" id=\"MathJax-Span-441\" 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-442\"><span style=\"display: inline-block; position: relative; width: 5.082em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.107em, -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-443\"><span class=\"mrow\" id=\"MathJax-Span-444\"><span class=\"mn\" id=\"MathJax-Span-445\" 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-452\"><span class=\"mrow\" id=\"MathJax-Span-453\"><span class=\"mn\" id=\"MathJax-Span-454\" 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-461\"><span class=\"mrow\" id=\"MathJax-Span-462\"><span class=\"mn\" id=\"MathJax-Span-463\" 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(2.425em, 1001.21em, 6.247em, -999.998em); top: -4.564em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-446\"><span class=\"mrow\" id=\"MathJax-Span-447\"><span class=\"mn\" id=\"MathJax-Span-448\" 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(3.171em, 1000.47em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-455\"><span class=\"mrow\" id=\"MathJax-Span-456\"><span class=\"mn\" id=\"MathJax-Span-457\" style=\"font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-464\"><span class=\"mrow\" id=\"MathJax-Span-465\"><span class=\"mo\" id=\"MathJax-Span-466\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-467\" 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: 4.569em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.425em, 1001.21em, 6.247em, -999.998em); top: -4.564em; left: 3.777em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-449\"><span class=\"mrow\" id=\"MathJax-Span-450\"><span class=\"mn\" id=\"MathJax-Span-451\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-458\"><span class=\"mrow\" id=\"MathJax-Span-459\"><span class=\"mn\" id=\"MathJax-Span-460\" 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: -2.561em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-468\"><span class=\"mrow\" id=\"MathJax-Span-469\"><span class=\"mo\" id=\"MathJax-Span-470\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-471\" 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: 4.569em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-472\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Size4; top: -2.84em; left: 0em;\">\u23a4<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Size4; top: -0.836em; left: 0em;\">\u23a6<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Size4; position: absolute; top: -1.815em; left: 0em;\">\u23a5<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-13\">\\left[\\begin{array}{ccc}  1 & 5 & 3 \\\\  2 & 4 & 0 \\\\  3 & -1 & -5  \\end{array}\\right]<\/script> and B = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-14-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-473\" 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-474\"><span class=\"mrow\" id=\"MathJax-Span-475\"><span class=\"mo\" id=\"MathJax-Span-476\" 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-477\"><span style=\"display: inline-block; position: relative; width: 4.289em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.06em, -999.998em); top: -4.471em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><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 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-488\"><span class=\"mrow\" id=\"MathJax-Span-489\"><span class=\"mn\" id=\"MathJax-Span-490\" 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: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-498\"><span class=\"mrow\" id=\"MathJax-Span-499\"><span class=\"mn\" id=\"MathJax-Span-500\" 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: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-481\"><span class=\"mrow\" id=\"MathJax-Span-482\"><span class=\"mo\" id=\"MathJax-Span-483\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-484\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-491\"><span class=\"mrow\" id=\"MathJax-Span-492\"><span class=\"mo\" id=\"MathJax-Span-493\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-494\" 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-501\"><span class=\"mrow\" id=\"MathJax-Span-502\"><span class=\"mn\" id=\"MathJax-Span-503\" 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, 6.107em, -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-485\"><span class=\"mrow\" id=\"MathJax-Span-486\"><span class=\"mn\" id=\"MathJax-Span-487\" 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: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-495\"><span class=\"mrow\" id=\"MathJax-Span-496\"><span class=\"mn\" id=\"MathJax-Span-497\" style=\"font-family: MathJax_Main;\">5<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-504\"><span class=\"mrow\" id=\"MathJax-Span-505\"><span class=\"mn\" id=\"MathJax-Span-506\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-507\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Size4; top: -2.84em; left: 0em;\">\u23a4<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Size4; top: -0.836em; left: 0em;\">\u23a6<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Size4; position: absolute; top: -1.815em; left: 0em;\">\u23a5<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-14\">\\left[\\begin{array}{ccc}  2 & -1 & 0 \\\\  0 & -2 & 5 \\\\  1 & 2 & 0  \\end{array}\\right]<\/script>, then find 3A \u2013 4B\u2019.<\/h3>\n\n<p>Answer:<br> \n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-5749\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3c-6.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(c) 6\" width=\"299\" height=\"272\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(c) 6\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(c)\"><\/p><p>Question 3.<br> If A = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-15-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-508\" 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, 1004.89em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-509\"><span class=\"mrow\" id=\"MathJax-Span-510\"><span class=\"mo\" id=\"MathJax-Span-511\" 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-512\"><span style=\"display: inline-block; position: relative; width: 3.544em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.379em, 1001.21em, 6.153em, -999.998em); top: -4.518em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-513\"><span class=\"mrow\" id=\"MathJax-Span-514\"><span class=\"mn\" id=\"MathJax-Span-515\" style=\"font-family: MathJax_Main;\">7<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-520\"><span class=\"mrow\" id=\"MathJax-Span-521\"><span class=\"mo\" id=\"MathJax-Span-522\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-523\" 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-527\"><span class=\"mrow\" id=\"MathJax-Span-528\"><span class=\"mn\" id=\"MathJax-Span-529\" 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: 4.522em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1001.21em, 6.107em, -999.998em); top: -4.471em; left: 2.286em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-516\"><span class=\"mrow\" id=\"MathJax-Span-517\"><span class=\"mo\" id=\"MathJax-Span-518\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-519\" 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-524\"><span class=\"mrow\" id=\"MathJax-Span-525\"><span class=\"mn\" id=\"MathJax-Span-526\" 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-530\"><span class=\"mrow\" id=\"MathJax-Span-531\"><span class=\"mn\" id=\"MathJax-Span-532\" 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-533\" 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}{rr}  7 & -2 \\\\  -1 & 2 \\\\  5 & 3  \\end{array}\\right]<\/script> and B = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-16-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-534\" 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, 1004.89em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-535\"><span class=\"mrow\" id=\"MathJax-Span-536\"><span class=\"mo\" id=\"MathJax-Span-537\" 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-538\"><span style=\"display: inline-block; position: relative; width: 3.544em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.425em, 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, 1001.21em, 4.243em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-539\"><span class=\"mrow\" id=\"MathJax-Span-540\"><span class=\"mo\" id=\"MathJax-Span-541\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-542\" 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-547\"><span class=\"mrow\" id=\"MathJax-Span-548\"><span class=\"mn\" id=\"MathJax-Span-549\" style=\"font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-553\"><span class=\"mrow\" id=\"MathJax-Span-554\"><span class=\"mo\" id=\"MathJax-Span-555\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-556\" 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: 4.569em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1001.21em, 6.107em, -999.998em); top: -4.471em; left: 2.286em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-543\"><span class=\"mrow\" id=\"MathJax-Span-544\"><span class=\"mo\" id=\"MathJax-Span-545\" style=\"font-family: MathJax_Main;\">\u2212<\/span><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, 1000.47em, 4.15em, -999.998em); top: -3.959em; right: 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;\">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-557\"><span class=\"mrow\" id=\"MathJax-Span-558\"><span class=\"mn\" id=\"MathJax-Span-559\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-560\" 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}{rr}  -2 & -1 \\\\  4 & 2 \\\\  -1 & 0  \\end{array}\\right]<\/script> then find AB\u2019 and BA\u2019.<\/h3>\n\n<p>Answer:<br> \n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-5747\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3c-7.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(c) 7\" width=\"336\" height=\"623\" \nsizes=\"auto, (max-width: 336px) 100vw, 336px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(c) 7\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(c)\"><\/p>\n\n<h3>Question 4.<br> For any square matrix A; show that A A\u2019 is symmetric. (March 2015-A.P)<\/h3>\n\n<p>Answer:<br> By definition a matrix is said to be symmetric if A\u2019 = A.<br> \u2234(A A\u2019)\u2019 = (A\u2019)\u2019 A\u2019 = A A\u2019<br> [(\u2235 (AB)\u2019 = B\u2019A\u2019 and (A\u2019)\u2019 = A]<br> Hence AA\u2019 is a symmetric matrix.<\/p><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Question 1. If A = [2\u221210115] and B = [\u221210110\u22122] then find (AB\u2019)\u2019 Answer: We have (AB)\u2019 = B\u2019A\u2019 and (AB\u2019)\u2019 = (B\u2019)\u2019 A\u2019 = BA\u2019 (\u2235 (B )\u2019 = B) Question 2. If A = \u23a1\u23a3\u23a2\u221225\u22121104\u23a4\u23a6\u23a5 and B = [\u2212243012] then find 2A + B\u2019 and 3B\u2019 \u2013 A. Answer: Question 3. If A [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":2695,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[1544,15,38],"tags":[1615,1574,1612,1616,1611,1610,1609,1613,1614],"class_list":{"0":"post-2689","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-study-material-tg-inter","8":"category-telangana","9":"category-tg-inter","10":"tag-bie-telangana-maths-matrices-solutions","11":"tag-inter-1st-year-maths-1a-matrices-important-questions","12":"tag-telangana-intermediate-maths-1a-chapter-3-solutions","13":"tag-ts-inter-1a-maths-textbook-solutions","14":"tag-ts-inter-1st-year-maths-1a-matrices-solutions","15":"tag-ts-inter-1st-year-matrices-solutions-pdf","16":"tag-ts-inter-maths-1a-exercise-3-solutions","17":"tag-ts-inter-maths-1a-matrices-exercise-3-answers","18":"tag-ts-intermediate-1st-year-maths-1a-guide"},"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\r\n<title>TS Inter 1st Year Maths 1A Matrices Solutions Exercise 3(c)<\/title>\r\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\r\n<link rel=\"canonical\" href=\"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3c\/\" \/>\r\n<meta property=\"og:locale\" content=\"en_US\" \/>\r\n<meta property=\"og:type\" content=\"article\" \/>\r\n<meta property=\"og:title\" content=\"TS Inter 1st Year Maths 1A Matrices Solutions Exercise 3(c)\" \/>\r\n<meta property=\"og:description\" content=\"Question 1. If A = [2\u221210115] and B = [\u221210110\u22122] then find (AB\u2019)\u2019 Answer: We have (AB)\u2019 = B\u2019A\u2019 and (AB\u2019)\u2019 = (B\u2019)\u2019 A\u2019 = BA\u2019 (\u2235 (B )\u2019 = B) Question 2. If A = \u23a1\u23a3\u23a2\u221225\u22121104\u23a4\u23a6\u23a5 and B = [\u2212243012] then find 2A + B\u2019 and 3B\u2019 \u2013 A. Answer: Question 3. If A [&hellip;]\" \/>\r\n<meta property=\"og:url\" content=\"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3c\/\" \/>\r\n<meta property=\"og:site_name\" content=\"Manabadi Boards\" \/>\r\n<meta property=\"article:published_time\" content=\"2026-01-08T07:18:13+00:00\" \/>\r\n<meta property=\"article:modified_time\" content=\"2026-01-08T08:25:52+00:00\" \/>\r\n<meta property=\"og:image\" content=\"https:\/\/boardscdn.manabadi.co.in\/wp-content\/uploads\/2026\/01\/07114619\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3c.jpg\" \/>\r\n\t<meta property=\"og:image:width\" content=\"900\" \/>\r\n\t<meta property=\"og:image:height\" content=\"600\" \/>\r\n\t<meta property=\"og:image:type\" content=\"image\/jpeg\" \/>\r\n<meta name=\"author\" content=\"Junaid\" \/>\r\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\r\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Junaid\" 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If A = [2\u221210115] and B = [\u221210110\u22122] then find (AB\u2019)\u2019 Answer: We have (AB)\u2019 = B\u2019A\u2019 and (AB\u2019)\u2019 = (B\u2019)\u2019 A\u2019 = BA\u2019 (\u2235 (B )\u2019 = B) Question 2. If A = \u23a1\u23a3\u23a2\u221225\u22121104\u23a4\u23a6\u23a5 and B = [\u2212243012] then find 2A + B\u2019 and 3B\u2019 \u2013 A. Answer: Question 3. If A [&hellip;]","og_url":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3c\/","og_site_name":"Manabadi Boards","article_published_time":"2026-01-08T07:18:13+00:00","article_modified_time":"2026-01-08T08:25:52+00:00","og_image":[{"width":900,"height":600,"url":"https:\/\/boardscdn.manabadi.co.in\/wp-content\/uploads\/2026\/01\/07114619\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3c.jpg","type":"image\/jpeg"}],"author":"Junaid","twitter_card":"summary_large_image","twitter_misc":{"Written by":"Junaid","Est. reading time":"4 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