{"id":2707,"date":"2026-01-10T12:45:07","date_gmt":"2026-01-10T07:15:07","guid":{"rendered":"https:\/\/www.manabadi.co.in\/boards\/?p=2707"},"modified":"2026-01-10T12:45:09","modified_gmt":"2026-01-10T07:15:09","slug":"ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3e","status":"publish","type":"post","link":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3e\/","title":{"rendered":"TS Inter 1st Year Maths 1A Matrices Solutions Exercise 3(e)"},"content":{"rendered":"\n<h2>TS Inter 1st Year Maths 1A Matrices Solutions Exercise 3(e)<\/h2>\n\n<h3>I.<br> Question 1.<br> Find the adjoint and inverse of the following matrices. (March 2002)<\/h3>\n\n<p>i) <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-1-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-1\" style=\"width: 4.942em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.243em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.773em, 1004.01em, 4.429em, -999.998em); top: -3.353em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-2\"><span class=\"mrow\" id=\"MathJax-Span-3\"><span class=\"mo\" id=\"MathJax-Span-4\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-5\"><span style=\"display: inline-block; position: relative; width: 2.798em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.565em, 1000.47em, 4.895em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -4.658em; 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(3.171em, 1000.47em, 4.15em, -999.998em); top: -3.26em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-13\"><span class=\"mrow\" id=\"MathJax-Span-14\"><span class=\"mn\" id=\"MathJax-Span-15\" style=\"font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1001.26em, 4.942em, -999.998em); top: -4.005em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.26em, 4.243em, -999.998em); top: -4.658em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-9\"><span class=\"mrow\" id=\"MathJax-Span-10\"><span class=\"mo\" id=\"MathJax-Span-11\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-12\" 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-16\"><span class=\"mrow\" id=\"MathJax-Span-17\"><span class=\"mn\" id=\"MathJax-Span-18\" style=\"font-family: MathJax_Main;\">6<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-19\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">]<\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 3.357em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.132em; border-left: 0px solid; width: 0px; height: 2.922em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-1\">\\left[\\begin{array}{rr} 2 & -3 \\\\ 4 & 6 \\end{array}\\right]<\/script><br>\n\nAnswer:<br> If A = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-2-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-20\" style=\"width: 4.056em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.497em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.773em, 1003.26em, 4.429em, -999.998em); top: -3.353em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-21\"><span class=\"mrow\" id=\"MathJax-Span-22\"><span class=\"mo\" id=\"MathJax-Span-23\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-24\"><span style=\"display: inline-block; position: relative; width: 2.053em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.798em, 1000.51em, 4.895em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-25\"><span class=\"mrow\" id=\"MathJax-Span-26\"><span class=\"mi\" id=\"MathJax-Span-27\" style=\"font-family: MathJax_Math-italic;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.42em, 4.15em, -999.998em); top: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-31\"><span class=\"mrow\" id=\"MathJax-Span-32\"><span class=\"mi\" id=\"MathJax-Span-33\" style=\"font-family: MathJax_Math-italic;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.519em, 1000.51em, 4.895em, -999.998em); top: -4.005em; left: 1.54em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.42em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-28\"><span class=\"mrow\" id=\"MathJax-Span-29\"><span class=\"mi\" id=\"MathJax-Span-30\" style=\"font-family: MathJax_Math-italic;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.51em, 4.15em, -999.998em); top: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-34\"><span class=\"mrow\" id=\"MathJax-Span-35\"><span class=\"mi\" id=\"MathJax-Span-36\" style=\"font-family: MathJax_Math-italic;\">d<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-37\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">]<\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 3.357em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.132em; border-left: 0px solid; width: 0px; height: 2.922em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-2\">\\left[\\begin{array}{ll} a & b \\\\ c & d \\end{array}\\right]<\/script> then adj 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-38\" 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-39\"><span class=\"mrow\" id=\"MathJax-Span-40\"><span class=\"mo\" id=\"MathJax-Span-41\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-42\"><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(2.519em, 1001.21em, 4.988em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 1.214em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.51em, 4.15em, -999.998em); top: -4.658em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-43\"><span class=\"mrow\" id=\"MathJax-Span-44\"><span class=\"texatom\" id=\"MathJax-Span-45\"><span class=\"mrow\" id=\"MathJax-Span-46\"><span class=\"mi\" id=\"MathJax-Span-47\" style=\"font-family: MathJax_Main;\">d<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.264em, 1001.21em, 4.243em, -999.998em); top: -3.26em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-54\"><span class=\"mrow\" id=\"MathJax-Span-55\"><span class=\"mo\" id=\"MathJax-Span-56\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"texatom\" id=\"MathJax-Span-57\"><span class=\"mrow\" id=\"MathJax-Span-58\"><span class=\"mi\" id=\"MathJax-Span-59\" style=\"font-family: MathJax_Main;\">c<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.519em, 1001.31em, 4.895em, -999.998em); top: -4.005em; left: 2.239em;\"><span style=\"display: inline-block; position: relative; width: 1.354em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.31em, 4.243em, -999.998em); top: -4.658em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-48\"><span class=\"mrow\" id=\"MathJax-Span-49\"><span class=\"mo\" id=\"MathJax-Span-50\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"texatom\" id=\"MathJax-Span-51\"><span class=\"mrow\" id=\"MathJax-Span-52\"><span class=\"mi\" id=\"MathJax-Span-53\" style=\"font-family: MathJax_Main;\">b<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.51em, 4.15em, -999.998em); top: -3.26em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-60\"><span class=\"mrow\" id=\"MathJax-Span-61\"><span class=\"texatom\" id=\"MathJax-Span-62\"><span class=\"mrow\" id=\"MathJax-Span-63\"><span class=\"mi\" id=\"MathJax-Span-64\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-65\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">]<\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 3.357em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.132em; border-left: 0px solid; width: 0px; height: 2.922em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-3\">\\left[\\begin{array}{rr} \\mathrm{d} & -\\mathrm{b} \\\\ -\\mathrm{c} & \\mathrm{a} \\end{array}\\right]<\/script>\n\n<br> <img fetchpriority=\"high\" decoding=\"async\" class=\"alignnone size-full wp-image-6223\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3e-1.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(e) 1\" width=\"284\" height=\"349\" sizes=\"(max-width: 284px) 100vw, 284px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(e) 1\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(e)\"><\/p>\n\n<p>ii) <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-4-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-66\" 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-67\"><span class=\"mrow\" id=\"MathJax-Span-68\"><span class=\"mo\" id=\"MathJax-Span-69\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-70\"><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, 1002.15em, 4.895em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 2.192em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1002.15em, 4.15em, -999.998em); top: -4.658em; left: 50%; margin-left: -1.069em;\"><span class=\"mtd\" id=\"MathJax-Span-71\"><span class=\"mrow\" id=\"MathJax-Span-72\"><span class=\"mi\" id=\"MathJax-Span-73\" style=\"font-family: MathJax_Main;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-74\"><\/span><span class=\"mi\" id=\"MathJax-Span-75\" 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, 1002.01em, 4.15em, -999.998em); top: -3.26em; left: 50%; margin-left: -1.023em;\"><span class=\"mtd\" id=\"MathJax-Span-82\"><span class=\"mrow\" id=\"MathJax-Span-83\"><span class=\"mi\" id=\"MathJax-Span-84\" style=\"font-family: MathJax_Main;\">sin<\/span><span class=\"mo\" id=\"MathJax-Span-85\"><\/span><span class=\"mi\" id=\"MathJax-Span-86\" 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, 1002.99em, 4.895em, -999.998em); top: -4.005em; left: 3.171em;\"><span style=\"display: inline-block; position: relative; width: 3.031em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1002.99em, 4.243em, -999.998em); top: -4.658em; left: 50%; margin-left: -1.489em;\"><span class=\"mtd\" id=\"MathJax-Span-76\"><span class=\"mrow\" id=\"MathJax-Span-77\"><span class=\"mo\" id=\"MathJax-Span-78\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mi\" id=\"MathJax-Span-79\" style=\"font-family: MathJax_Main; padding-left: 0.189em;\">sin<\/span><span class=\"mo\" id=\"MathJax-Span-80\"><\/span><span class=\"mi\" id=\"MathJax-Span-81\" 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-87\"><span class=\"mrow\" id=\"MathJax-Span-88\"><span class=\"mi\" id=\"MathJax-Span-89\" style=\"font-family: MathJax_Main;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-90\"><\/span><span class=\"mi\" id=\"MathJax-Span-91\" 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-92\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">]<\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 3.357em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.132em; border-left: 0px solid; width: 0px; height: 2.922em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-4\">\\left[\\begin{array}{cc} \\cos \\alpha & -\\sin \\alpha \\\\ \\sin \\alpha & \\cos \\alpha \\end{array}\\right]<\/script><br>\n\nAnswer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-6222\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3e-2.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(e) 2\" width=\"218\" height=\"223\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(e) 2\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(e)\"><\/p>\n\n<p>iii) Find the adjoint and inverse of the matrix <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-5-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-93\" style=\"width: 6.014em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 5.175em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1004.85em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-94\"><span class=\"mrow\" id=\"MathJax-Span-95\"><span class=\"mo\" id=\"MathJax-Span-96\" 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-97\"><span style=\"display: inline-block; position: relative; width: 3.497em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.107em, -999.998em); top: -4.471em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-98\"><span class=\"mrow\" id=\"MathJax-Span-99\"><span class=\"mn\" id=\"MathJax-Span-100\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-107\"><span class=\"mrow\" id=\"MathJax-Span-108\"><span class=\"mn\" id=\"MathJax-Span-109\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-116\"><span class=\"mrow\" id=\"MathJax-Span-117\"><span class=\"mn\" id=\"MathJax-Span-118\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.06em, -999.998em); top: -4.471em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-101\"><span class=\"mrow\" id=\"MathJax-Span-102\"><span class=\"mn\" id=\"MathJax-Span-103\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-110\"><span class=\"mrow\" id=\"MathJax-Span-111\"><span class=\"mn\" id=\"MathJax-Span-112\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-119\"><span class=\"mrow\" id=\"MathJax-Span-120\"><span class=\"mn\" id=\"MathJax-Span-121\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.06em, -999.998em); top: -4.471em; left: 2.985em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-104\"><span class=\"mrow\" id=\"MathJax-Span-105\"><span class=\"mn\" id=\"MathJax-Span-106\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-113\"><span class=\"mrow\" id=\"MathJax-Span-114\"><span class=\"mn\" id=\"MathJax-Span-115\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-122\"><span class=\"mrow\" id=\"MathJax-Span-123\"><span class=\"mn\" id=\"MathJax-Span-124\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-125\" 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-5\">\\left[\\begin{array}{lll} 1 & 0 & 2 \\\\ 2 & 1 & 0 \\\\ 3 & 2 & 1 \\end{array}\\right]<\/script>.<br> Answer:<br> Find cofactors of elements in the matrix as<br>\n\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-6221\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3e-3.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(e) 3\" width=\"305\" height=\"657\" sizes=\"auto, (max-width: 305px) 100vw, 305px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(e) 3\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(e)\"><\/p>\n\n<p>iv) <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-6-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-126\" style=\"width: 5.128em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.429em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1004.29em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-127\"><span class=\"mrow\" id=\"MathJax-Span-128\"><span class=\"mo\" id=\"MathJax-Span-129\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-130\"><span style=\"display: inline-block; position: relative; width: 3.497em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.06em, -999.998em); top: -4.471em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-131\"><span class=\"mrow\" id=\"MathJax-Span-132\"><span class=\"mn\" id=\"MathJax-Span-133\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-140\"><span class=\"mrow\" id=\"MathJax-Span-141\"><span class=\"mn\" id=\"MathJax-Span-142\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-149\"><span class=\"mrow\" id=\"MathJax-Span-150\"><span class=\"mn\" id=\"MathJax-Span-151\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.06em, -999.998em); top: -4.471em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-134\"><span class=\"mrow\" id=\"MathJax-Span-135\"><span class=\"mn\" id=\"MathJax-Span-136\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-143\"><span class=\"mrow\" id=\"MathJax-Span-144\"><span class=\"mn\" id=\"MathJax-Span-145\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-152\"><span class=\"mrow\" id=\"MathJax-Span-153\"><span class=\"mn\" id=\"MathJax-Span-154\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.06em, -999.998em); top: -4.471em; left: 2.985em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-137\"><span class=\"mrow\" id=\"MathJax-Span-138\"><span class=\"mn\" id=\"MathJax-Span-139\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-146\"><span class=\"mrow\" id=\"MathJax-Span-147\"><span class=\"mn\" id=\"MathJax-Span-148\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-155\"><span class=\"mrow\" id=\"MathJax-Span-156\"><span class=\"mn\" id=\"MathJax-Span-157\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-158\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-6\">\\left|\\begin{array}{lll} 2 & 1 & 2 \\\\ 1 & 0 & 1 \\\\ 2 & 2 & 1 \\end{array}\\right|<\/script><br>\n\nAnswer:<br>\n\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-6220\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3e-4.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(e) 4\" width=\"303\" height=\"680\" sizes=\"auto, (max-width: 303px) 100vw, 303px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(e) 4\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(e)\"><\/p>\n\n<h3>Question 2.<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-159\" style=\"width: 9.695em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 8.343em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.773em, 1008.11em, 4.429em, -999.998em); top: -3.353em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-160\"><span class=\"mrow\" id=\"MathJax-Span-161\"><span class=\"mo\" id=\"MathJax-Span-162\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-163\"><span style=\"display: inline-block; position: relative; width: 6.899em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.519em, 1003.26em, 4.988em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 3.311em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1002.57em, 4.243em, -999.998em); top: -4.658em; left: 50%; margin-left: -1.302em;\"><span class=\"mtd\" id=\"MathJax-Span-164\"><span class=\"mrow\" id=\"MathJax-Span-165\"><span class=\"texatom\" id=\"MathJax-Span-166\"><span class=\"mrow\" id=\"MathJax-Span-167\"><span class=\"mi\" id=\"MathJax-Span-168\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-169\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"texatom\" id=\"MathJax-Span-170\" style=\"padding-left: 0.235em;\"><span class=\"mrow\" id=\"MathJax-Span-171\"><span class=\"mi\" id=\"MathJax-Span-172\" style=\"font-family: MathJax_Main;\">i<\/span><span class=\"mi\" id=\"MathJax-Span-173\" style=\"font-family: MathJax_Main;\">b<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1003.26em, 4.243em, -999.998em); top: -3.26em; left: 50%; margin-left: -1.629em;\"><span class=\"mtd\" id=\"MathJax-Span-184\"><span class=\"mrow\" id=\"MathJax-Span-185\"><span class=\"mo\" id=\"MathJax-Span-186\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"texatom\" id=\"MathJax-Span-187\"><span class=\"mrow\" id=\"MathJax-Span-188\"><span class=\"mi\" id=\"MathJax-Span-189\" style=\"font-family: MathJax_Main;\">c<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-190\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"texatom\" id=\"MathJax-Span-191\" style=\"padding-left: 0.235em;\"><span class=\"mrow\" id=\"MathJax-Span-192\"><span class=\"mi\" id=\"MathJax-Span-193\" style=\"font-family: MathJax_Main;\">i<\/span><span class=\"mi\" id=\"MathJax-Span-194\" style=\"font-family: MathJax_Main;\">d<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.519em, 1002.57em, 4.988em, -999.998em); top: -4.005em; left: 4.289em;\"><span style=\"display: inline-block; position: relative; width: 2.565em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1002.52em, 4.243em, -999.998em); top: -4.658em; left: 50%; margin-left: -1.256em;\"><span class=\"mtd\" id=\"MathJax-Span-174\"><span class=\"mrow\" id=\"MathJax-Span-175\"><span class=\"texatom\" id=\"MathJax-Span-176\"><span class=\"mrow\" id=\"MathJax-Span-177\"><span class=\"mi\" id=\"MathJax-Span-178\" style=\"font-family: MathJax_Main;\">c<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-179\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"texatom\" id=\"MathJax-Span-180\" style=\"padding-left: 0.235em;\"><span class=\"mrow\" id=\"MathJax-Span-181\"><span class=\"mi\" id=\"MathJax-Span-182\" style=\"font-family: MathJax_Main;\">i<\/span><span class=\"mi\" id=\"MathJax-Span-183\" style=\"font-family: MathJax_Main;\">d<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1002.57em, 4.243em, -999.998em); top: -3.26em; left: 50%; margin-left: -1.302em;\"><span class=\"mtd\" id=\"MathJax-Span-195\"><span class=\"mrow\" id=\"MathJax-Span-196\"><span class=\"texatom\" id=\"MathJax-Span-197\"><span class=\"mrow\" id=\"MathJax-Span-198\"><span class=\"mi\" id=\"MathJax-Span-199\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-200\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"texatom\" id=\"MathJax-Span-201\" style=\"padding-left: 0.235em;\"><span class=\"mrow\" id=\"MathJax-Span-202\"><span class=\"mi\" id=\"MathJax-Span-203\" style=\"font-family: MathJax_Main;\">i<\/span><span class=\"mi\" id=\"MathJax-Span-204\" style=\"font-family: MathJax_Main;\">b<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-205\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">]<\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 3.357em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.132em; border-left: 0px solid; width: 0px; height: 2.922em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-7\">\\left[\\begin{array}{cc} \\mathrm{a}+\\mathrm{i b} & \\mathrm{c}+\\mathrm{i d} \\\\ -\\mathrm{c}+\\mathrm{i d} & \\mathrm{a}-\\mathrm{i b} \\end{array}\\right]<\/script>, a<sup>2<\/sup> + b<sup>2<\/sup> + c<sup>2<\/sup> + d<sup>2<\/sup> = 1<\/h3>\n\n<p>Answer:<br> det A = (a + ib) (a \u2013 ib) \u2013 (c + id) (- c + id)<br> = (a<sup>2<\/sup> \u2013 i<sup>2<\/sup> b<sup>2<\/sup>) \u2013 (- c<sup>2<\/sup> + i<sup>2<\/sup>d<sup>2<\/sup>)<br> = a<sup>2<\/sup> + b<sup>2<\/sup> + c<sup>2<\/sup> + d<sup>2<\/sup> (\u2235 i<sup>2<\/sup> = -1)<br> = 1<br> Adj A = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-8-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-206\" style=\"width: 9.695em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 8.343em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.773em, 1008.11em, 4.429em, -999.998em); top: -3.353em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-207\"><span class=\"mrow\" id=\"MathJax-Span-208\"><span class=\"mo\" id=\"MathJax-Span-209\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-210\"><span style=\"display: inline-block; position: relative; width: 6.899em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.519em, 1002.57em, 4.988em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 2.565em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1002.57em, 4.243em, -999.998em); top: -4.658em; left: 50%; margin-left: -1.256em;\"><span class=\"mtd\" id=\"MathJax-Span-211\"><span class=\"mrow\" id=\"MathJax-Span-212\"><span class=\"mi\" id=\"MathJax-Span-213\" style=\"font-family: MathJax_Math-italic;\">a<\/span><span class=\"mo\" id=\"MathJax-Span-214\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"mi\" id=\"MathJax-Span-215\" style=\"font-family: MathJax_Math-italic; padding-left: 0.235em;\">i<\/span><span class=\"mi\" id=\"MathJax-Span-216\" style=\"font-family: MathJax_Math-italic;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1002.57em, 4.243em, -999.998em); top: -3.26em; left: 50%; margin-left: -1.256em;\"><span class=\"mtd\" id=\"MathJax-Span-224\"><span class=\"mrow\" id=\"MathJax-Span-225\"><span class=\"mi\" id=\"MathJax-Span-226\" style=\"font-family: MathJax_Math-italic;\">c<\/span><span class=\"mo\" id=\"MathJax-Span-227\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"mi\" id=\"MathJax-Span-228\" style=\"font-family: MathJax_Math-italic; padding-left: 0.235em;\">i<\/span><span class=\"mi\" id=\"MathJax-Span-229\" style=\"font-family: MathJax_Math-italic;\">d<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.519em, 1003.31em, 4.988em, -999.998em); top: -4.005em; left: 3.544em;\"><span style=\"display: inline-block; position: relative; width: 3.311em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1003.31em, 4.243em, -999.998em); top: -4.658em; left: 50%; margin-left: -1.675em;\"><span class=\"mtd\" id=\"MathJax-Span-217\"><span class=\"mrow\" id=\"MathJax-Span-218\"><span class=\"mo\" id=\"MathJax-Span-219\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mi\" id=\"MathJax-Span-220\" style=\"font-family: MathJax_Math-italic;\">c<\/span><span class=\"mo\" id=\"MathJax-Span-221\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"mi\" id=\"MathJax-Span-222\" style=\"font-family: MathJax_Math-italic; padding-left: 0.235em;\">i<\/span><span class=\"mi\" id=\"MathJax-Span-223\" style=\"font-family: MathJax_Math-italic;\">d<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1002.57em, 4.243em, -999.998em); top: -3.26em; left: 50%; margin-left: -1.256em;\"><span class=\"mtd\" id=\"MathJax-Span-230\"><span class=\"mrow\" id=\"MathJax-Span-231\"><span class=\"mi\" id=\"MathJax-Span-232\" style=\"font-family: MathJax_Math-italic;\">a<\/span><span class=\"mo\" id=\"MathJax-Span-233\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mi\" id=\"MathJax-Span-234\" style=\"font-family: MathJax_Math-italic; padding-left: 0.235em;\">i<\/span><span class=\"mi\" id=\"MathJax-Span-235\" style=\"font-family: MathJax_Math-italic;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-236\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">]<\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 3.357em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.132em; border-left: 0px solid; width: 0px; height: 2.922em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-8\">\\left[\\begin{array}{cc} a-i b & -c-i d \\\\ c-i d & a+i b \\end{array}\\right]<\/script><br> A<sup>-1<\/sup> = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-9-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-237\" style=\"width: 13.749em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 11.838em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.773em, 1011.61em, 4.429em, -999.998em); top: -3.353em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-238\"><span class=\"mfrac\" id=\"MathJax-Span-239\"><span style=\"display: inline-block; position: relative; width: 1.866em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.357em, 1001.73em, 4.289em, -999.998em); top: -4.564em; left: 50%; margin-left: -0.836em;\"><span class=\"mrow\" id=\"MathJax-Span-240\"><span class=\"texatom\" id=\"MathJax-Span-241\"><span class=\"mrow\" id=\"MathJax-Span-242\"><span class=\"mi\" id=\"MathJax-Span-243\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">A<\/span><span class=\"mi\" id=\"MathJax-Span-244\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">d<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><span class=\"mi\" id=\"MathJax-Span-245\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">j<\/span><\/span><\/span><span class=\"mi\" id=\"MathJax-Span-246\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">A<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.49em, 4.15em, -999.998em); top: -3.586em; left: 50%; margin-left: -0.743em;\"><span class=\"mrow\" id=\"MathJax-Span-247\"><span class=\"texatom\" id=\"MathJax-Span-248\"><span class=\"mrow\" id=\"MathJax-Span-249\"><span class=\"mi\" id=\"MathJax-Span-250\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">d<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><span class=\"mi\" id=\"MathJax-Span-251\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">e<\/span><span class=\"mi\" id=\"MathJax-Span-252\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">t<\/span><\/span><\/span><span class=\"mi\" id=\"MathJax-Span-253\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">A<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.87em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.866em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-254\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"mrow\" id=\"MathJax-Span-255\" style=\"padding-left: 0.282em;\"><span class=\"mo\" id=\"MathJax-Span-256\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-257\"><span style=\"display: inline-block; position: relative; width: 6.899em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.519em, 1002.57em, 4.988em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 2.565em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1002.57em, 4.243em, -999.998em); top: -4.658em; left: 50%; margin-left: -1.256em;\"><span class=\"mtd\" id=\"MathJax-Span-258\"><span class=\"mrow\" id=\"MathJax-Span-259\"><span class=\"mi\" id=\"MathJax-Span-260\" style=\"font-family: MathJax_Math-italic;\">a<\/span><span class=\"mo\" id=\"MathJax-Span-261\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"mi\" id=\"MathJax-Span-262\" style=\"font-family: MathJax_Math-italic; padding-left: 0.235em;\">i<\/span><span class=\"mi\" id=\"MathJax-Span-263\" style=\"font-family: MathJax_Math-italic;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1002.57em, 4.243em, -999.998em); top: -3.26em; left: 50%; margin-left: -1.256em;\"><span class=\"mtd\" id=\"MathJax-Span-271\"><span class=\"mrow\" id=\"MathJax-Span-272\"><span class=\"mi\" id=\"MathJax-Span-273\" style=\"font-family: MathJax_Math-italic;\">c<\/span><span class=\"mo\" id=\"MathJax-Span-274\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"mi\" id=\"MathJax-Span-275\" style=\"font-family: MathJax_Math-italic; padding-left: 0.235em;\">i<\/span><span class=\"mi\" id=\"MathJax-Span-276\" style=\"font-family: MathJax_Math-italic;\">d<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.519em, 1003.31em, 4.988em, -999.998em); top: -4.005em; left: 3.544em;\"><span style=\"display: inline-block; position: relative; width: 3.311em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1003.31em, 4.243em, -999.998em); top: -4.658em; left: 50%; margin-left: -1.675em;\"><span class=\"mtd\" id=\"MathJax-Span-264\"><span class=\"mrow\" id=\"MathJax-Span-265\"><span class=\"mo\" id=\"MathJax-Span-266\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mi\" id=\"MathJax-Span-267\" style=\"font-family: MathJax_Math-italic;\">c<\/span><span class=\"mo\" id=\"MathJax-Span-268\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"mi\" id=\"MathJax-Span-269\" style=\"font-family: MathJax_Math-italic; padding-left: 0.235em;\">i<\/span><span class=\"mi\" id=\"MathJax-Span-270\" style=\"font-family: MathJax_Math-italic;\">d<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1002.57em, 4.243em, -999.998em); top: -3.26em; left: 50%; margin-left: -1.256em;\"><span class=\"mtd\" id=\"MathJax-Span-277\"><span class=\"mrow\" id=\"MathJax-Span-278\"><span class=\"mi\" id=\"MathJax-Span-279\" style=\"font-family: MathJax_Math-italic;\">a<\/span><span class=\"mo\" id=\"MathJax-Span-280\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mi\" id=\"MathJax-Span-281\" style=\"font-family: MathJax_Math-italic; padding-left: 0.235em;\">i<\/span><span class=\"mi\" id=\"MathJax-Span-282\" style=\"font-family: MathJax_Math-italic;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-283\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">]<\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 3.357em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.132em; border-left: 0px solid; width: 0px; height: 2.922em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-9\">\\frac{{Adj} A}{{det} A}=\\left[\\begin{array}{cc} a-i b & -c-i d \\\\ c-i d & a+i b \\end{array}\\right]<\/script><\/p>\n\n<h3>Question 3.<br> If 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-284\" 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-285\"><span class=\"mrow\" id=\"MathJax-Span-286\"><span class=\"mo\" id=\"MathJax-Span-287\" 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-288\"><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, 1001.21em, 6.247em, -999.998em); top: -4.564em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-289\"><span class=\"mrow\" id=\"MathJax-Span-290\"><span class=\"mn\" id=\"MathJax-Span-291\" 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-299\"><span class=\"mrow\" id=\"MathJax-Span-300\"><span class=\"mn\" id=\"MathJax-Span-301\" 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-309\"><span class=\"mrow\" id=\"MathJax-Span-310\"><span class=\"mo\" id=\"MathJax-Span-311\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-312\" 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, 6.06em, -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; 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;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-302\"><span class=\"mrow\" id=\"MathJax-Span-303\"><span class=\"mo\" id=\"MathJax-Span-304\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-305\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-313\"><span class=\"mrow\" id=\"MathJax-Span-314\"><span class=\"mn\" id=\"MathJax-Span-315\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.06em, -999.998em); top: -4.471em; left: 4.569em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-296\"><span class=\"mrow\" id=\"MathJax-Span-297\"><span class=\"mn\" id=\"MathJax-Span-298\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-306\"><span class=\"mrow\" id=\"MathJax-Span-307\"><span class=\"mn\" id=\"MathJax-Span-308\" style=\"font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-316\"><span class=\"mrow\" id=\"MathJax-Span-317\"><span class=\"mn\" id=\"MathJax-Span-318\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-319\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Size4; top: -2.84em; left: 0em;\">\u23a4<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Size4; top: -0.836em; left: 0em;\">\u23a6<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Size4; position: absolute; top: -1.815em; left: 0em;\">\u23a5<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-10\">\\left[\\begin{array}{ccc} 1 & -2 & 3 \\\\ 0 & -1 & 4 \\\\ -2 & 2 & 1 \\end{array}\\right]<\/script>, then find (A\u2019)<sup>-1<\/sup>. (Board Model Paper)<\/h3>\n\n<p>Answer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-6219\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3e-5.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(e) 5\" width=\"298\" height=\"760\" sizes=\"auto, (max-width: 298px) 100vw, 298px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(e) 5\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(e)\"><\/p>\n\n<h3>Question 4.<br> If A = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-11-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-320\" style=\"width: 8.716em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 7.505em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1007.18em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-321\"><span class=\"mrow\" id=\"MathJax-Span-322\"><span class=\"mo\" id=\"MathJax-Span-323\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Size4; top: -2.84em; left: 0em;\">\u23a1<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Size4; top: -0.836em; left: 0em;\">\u23a3<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Size4; position: absolute; top: -1.815em; left: 0em;\">\u23a2<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-324\"><span style=\"display: inline-block; position: relative; width: 5.827em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1001.21em, 6.06em, -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; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-325\"><span class=\"mrow\" id=\"MathJax-Span-326\"><span class=\"mo\" id=\"MathJax-Span-327\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-328\" 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-337\"><span class=\"mrow\" id=\"MathJax-Span-338\"><span class=\"mn\" id=\"MathJax-Span-339\" 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-347\"><span class=\"mrow\" id=\"MathJax-Span-348\"><span class=\"mn\" id=\"MathJax-Span-349\" 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(2.425em, 1001.21em, 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, 1001.21em, 4.243em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-329\"><span class=\"mrow\" id=\"MathJax-Span-330\"><span class=\"mo\" id=\"MathJax-Span-331\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-332\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-340\"><span class=\"mrow\" id=\"MathJax-Span-341\"><span class=\"mn\" id=\"MathJax-Span-342\" 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: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-350\"><span class=\"mrow\" id=\"MathJax-Span-351\"><span class=\"mo\" id=\"MathJax-Span-352\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-353\" 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, 6.06em, -999.998em); top: -4.471em; left: 4.569em;\"><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-333\"><span class=\"mrow\" id=\"MathJax-Span-334\"><span class=\"mo\" id=\"MathJax-Span-335\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-336\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-343\"><span class=\"mrow\" id=\"MathJax-Span-344\"><span class=\"mo\" id=\"MathJax-Span-345\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-346\" 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; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-354\"><span class=\"mrow\" id=\"MathJax-Span-355\"><span class=\"mn\" id=\"MathJax-Span-356\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-357\" 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-11\">\\left[\\begin{array}{rrr} -1 & -2 & -2 \\\\ 2 & 1 & -2 \\\\ 2 & -2 & 1 \\end{array}\\right]<\/script>, then show that the adjoint of A = 3A, find A<sup>-1<\/sup><\/h3>\n\n<p>Answer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-6218\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3e-6.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(e) 6\" width=\"327\" height=\"991\" sizes=\"auto, (max-width: 327px) 100vw, 327px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(e) 6\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(e)\"><\/p>\n\n \n\n<h3>Question 5.<br> If abc \u2260 0; find the inverse of <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-12-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-358\" style=\"width: 6.014em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 5.175em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1004.85em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-359\"><span class=\"mrow\" id=\"MathJax-Span-360\"><span class=\"mo\" id=\"MathJax-Span-361\" 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-362\"><span style=\"display: inline-block; position: relative; width: 3.497em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.51em, 5.92em, -999.998em); top: -4.285em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.51em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-363\"><span class=\"mrow\" id=\"MathJax-Span-364\"><span class=\"texatom\" id=\"MathJax-Span-365\"><span class=\"mrow\" id=\"MathJax-Span-366\"><span class=\"mi\" id=\"MathJax-Span-367\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-374\"><span class=\"mrow\" id=\"MathJax-Span-375\"><span class=\"mn\" id=\"MathJax-Span-376\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-383\"><span class=\"mrow\" id=\"MathJax-Span-384\"><span class=\"mn\" id=\"MathJax-Span-385\" 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: 4.289em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.107em, -999.998em); top: -4.471em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-368\"><span class=\"mrow\" id=\"MathJax-Span-369\"><span class=\"mn\" id=\"MathJax-Span-370\" 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, 1000.42em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-377\"><span class=\"mrow\" id=\"MathJax-Span-378\"><span class=\"mi\" id=\"MathJax-Span-379\" style=\"font-family: MathJax_Math-italic;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; left: 0em;\"><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;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.06em, -999.998em); top: -4.471em; left: 2.985em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-371\"><span class=\"mrow\" id=\"MathJax-Span-372\"><span class=\"mn\" id=\"MathJax-Span-373\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-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(3.451em, 1000.42em, 4.15em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-389\"><span class=\"mrow\" id=\"MathJax-Span-390\"><span class=\"mi\" id=\"MathJax-Span-391\" style=\"font-family: MathJax_Math-italic;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-392\" 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-12\">\\left[\\begin{array}{lll} \\mathrm{a} & 0 & 0 \\\\ 0 & b & 0 \\\\ 0 & 0 & c \\end{array}\\right]<\/script> (May 2006)<\/h3>\n\n<p>Answer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-6217\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3e-7.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(e) 7\" width=\"326\" height=\"809\" sizes=\"auto, (max-width: 326px) 100vw, 326px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(e) 7\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(e)\"><\/p>\n\n<h3>II.<br> Question 1.<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-393\" style=\"width: 12.118em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 10.44em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1010.11em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-394\"><span class=\"mrow\" id=\"MathJax-Span-395\"><span class=\"mo\" id=\"MathJax-Span-396\" 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-397\"><span style=\"display: inline-block; position: relative; width: 8.763em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.379em, 1002.24em, 6.247em, -999.998em); top: -4.564em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 2.239em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1002.24em, 4.243em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-398\"><span class=\"mrow\" id=\"MathJax-Span-399\"><span class=\"texatom\" id=\"MathJax-Span-400\"><span class=\"mrow\" id=\"MathJax-Span-401\"><span class=\"mi\" id=\"MathJax-Span-402\" style=\"font-family: MathJax_Main;\">b<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-403\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"texatom\" id=\"MathJax-Span-404\" style=\"padding-left: 0.235em;\"><span class=\"mrow\" id=\"MathJax-Span-405\"><span class=\"mi\" id=\"MathJax-Span-406\" style=\"font-family: MathJax_Main;\">c<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1002.24em, 4.243em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-425\"><span class=\"mrow\" id=\"MathJax-Span-426\"><span class=\"texatom\" id=\"MathJax-Span-427\"><span class=\"mrow\" id=\"MathJax-Span-428\"><span class=\"mi\" id=\"MathJax-Span-429\" style=\"font-family: MathJax_Main;\">c<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-430\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"texatom\" id=\"MathJax-Span-431\" style=\"padding-left: 0.235em;\"><span class=\"mrow\" id=\"MathJax-Span-432\"><span class=\"mi\" id=\"MathJax-Span-433\" style=\"font-family: MathJax_Main;\">b<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1002.24em, 4.243em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-452\"><span class=\"mrow\" id=\"MathJax-Span-453\"><span class=\"texatom\" id=\"MathJax-Span-454\"><span class=\"mrow\" id=\"MathJax-Span-455\"><span class=\"mi\" id=\"MathJax-Span-456\" style=\"font-family: MathJax_Main;\">b<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-457\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"texatom\" id=\"MathJax-Span-458\" style=\"padding-left: 0.235em;\"><span class=\"mrow\" id=\"MathJax-Span-459\"><span class=\"mi\" id=\"MathJax-Span-460\" style=\"font-family: MathJax_Main;\">c<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.425em, 1002.19em, 6.153em, -999.998em); top: -4.471em; left: 3.264em;\"><span style=\"display: inline-block; position: relative; width: 2.192em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.264em, 1002.19em, 4.243em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-407\"><span class=\"mrow\" id=\"MathJax-Span-408\"><span class=\"texatom\" id=\"MathJax-Span-409\"><span class=\"mrow\" id=\"MathJax-Span-410\"><span class=\"mi\" id=\"MathJax-Span-411\" style=\"font-family: MathJax_Main;\">c<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-412\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"texatom\" id=\"MathJax-Span-413\" style=\"padding-left: 0.235em;\"><span class=\"mrow\" id=\"MathJax-Span-414\"><span class=\"mi\" id=\"MathJax-Span-415\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.264em, 1002.19em, 4.243em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-434\"><span class=\"mrow\" id=\"MathJax-Span-435\"><span class=\"texatom\" id=\"MathJax-Span-436\"><span class=\"mrow\" id=\"MathJax-Span-437\"><span class=\"mi\" id=\"MathJax-Span-438\" style=\"font-family: MathJax_Main;\">c<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-439\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"texatom\" id=\"MathJax-Span-440\" style=\"padding-left: 0.235em;\"><span class=\"mrow\" id=\"MathJax-Span-441\"><span class=\"mi\" id=\"MathJax-Span-442\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.264em, 1002.15em, 4.243em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-461\"><span class=\"mrow\" id=\"MathJax-Span-462\"><span class=\"texatom\" id=\"MathJax-Span-463\"><span class=\"mrow\" id=\"MathJax-Span-464\"><span class=\"mi\" id=\"MathJax-Span-465\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-466\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"texatom\" id=\"MathJax-Span-467\" style=\"padding-left: 0.235em;\"><span class=\"mrow\" id=\"MathJax-Span-468\"><span class=\"mi\" id=\"MathJax-Span-469\" style=\"font-family: MathJax_Main;\">c<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.379em, 1002.29em, 6.247em, -999.998em); top: -4.564em; left: 6.433em;\"><span style=\"display: inline-block; position: relative; width: 2.286em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1002.29em, 4.243em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-416\"><span class=\"mrow\" id=\"MathJax-Span-417\"><span class=\"texatom\" id=\"MathJax-Span-418\"><span class=\"mrow\" id=\"MathJax-Span-419\"><span class=\"mi\" id=\"MathJax-Span-420\" style=\"font-family: MathJax_Main;\">b<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-421\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"texatom\" id=\"MathJax-Span-422\" style=\"padding-left: 0.235em;\"><span class=\"mrow\" id=\"MathJax-Span-423\"><span class=\"mi\" id=\"MathJax-Span-424\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1002.29em, 4.243em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-443\"><span class=\"mrow\" id=\"MathJax-Span-444\"><span class=\"texatom\" id=\"MathJax-Span-445\"><span class=\"mrow\" id=\"MathJax-Span-446\"><span class=\"mi\" id=\"MathJax-Span-447\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-448\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"texatom\" id=\"MathJax-Span-449\" style=\"padding-left: 0.235em;\"><span class=\"mrow\" id=\"MathJax-Span-450\"><span class=\"mi\" id=\"MathJax-Span-451\" style=\"font-family: MathJax_Main;\">b<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1002.29em, 4.243em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-470\"><span class=\"mrow\" id=\"MathJax-Span-471\"><span class=\"texatom\" id=\"MathJax-Span-472\"><span class=\"mrow\" id=\"MathJax-Span-473\"><span class=\"mi\" id=\"MathJax-Span-474\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-475\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"texatom\" id=\"MathJax-Span-476\" style=\"padding-left: 0.235em;\"><span class=\"mrow\" id=\"MathJax-Span-477\"><span class=\"mi\" id=\"MathJax-Span-478\" style=\"font-family: MathJax_Main;\">b<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-479\" 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}{lll} \\mathrm{b}+\\mathrm{c} & \\mathrm{c}-\\mathrm{a} & \\mathrm{b}-\\mathrm{a} \\\\ \\mathrm{c}-\\mathrm{b} & \\mathrm{c}+\\mathrm{a} & \\mathrm{a}-\\mathrm{b} \\\\ \\mathrm{b}-\\mathrm{c} & \\mathrm{a}-\\mathrm{c} & \\mathrm{a}+\\mathrm{b} \\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-480\" style=\"width: 13.003em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 11.186em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1010.86em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-481\"><span class=\"mfrac\" id=\"MathJax-Span-482\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.28em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-483\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-484\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mrow\" id=\"MathJax-Span-485\"><span class=\"mo\" id=\"MathJax-Span-486\" 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-487\"><span style=\"display: inline-block; position: relative; width: 8.763em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.379em, 1002.24em, 6.247em, -999.998em); top: -4.564em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 2.239em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1002.24em, 4.243em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-488\"><span class=\"mrow\" id=\"MathJax-Span-489\"><span class=\"texatom\" id=\"MathJax-Span-490\"><span class=\"mrow\" id=\"MathJax-Span-491\"><span class=\"mi\" id=\"MathJax-Span-492\" style=\"font-family: MathJax_Main;\">b<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-493\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"texatom\" id=\"MathJax-Span-494\" style=\"padding-left: 0.235em;\"><span class=\"mrow\" id=\"MathJax-Span-495\"><span class=\"mi\" id=\"MathJax-Span-496\" style=\"font-family: MathJax_Main;\">c<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1002.24em, 4.243em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-515\"><span class=\"mrow\" id=\"MathJax-Span-516\"><span class=\"texatom\" id=\"MathJax-Span-517\"><span class=\"mrow\" id=\"MathJax-Span-518\"><span class=\"mi\" id=\"MathJax-Span-519\" style=\"font-family: MathJax_Main;\">c<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-520\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"texatom\" id=\"MathJax-Span-521\" style=\"padding-left: 0.235em;\"><span class=\"mrow\" id=\"MathJax-Span-522\"><span class=\"mi\" id=\"MathJax-Span-523\" style=\"font-family: MathJax_Main;\">b<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1002.24em, 4.243em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-542\"><span class=\"mrow\" id=\"MathJax-Span-543\"><span class=\"texatom\" id=\"MathJax-Span-544\"><span class=\"mrow\" id=\"MathJax-Span-545\"><span class=\"mi\" id=\"MathJax-Span-546\" style=\"font-family: MathJax_Main;\">b<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-547\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"texatom\" id=\"MathJax-Span-548\" style=\"padding-left: 0.235em;\"><span class=\"mrow\" id=\"MathJax-Span-549\"><span class=\"mi\" id=\"MathJax-Span-550\" style=\"font-family: MathJax_Main;\">c<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.425em, 1002.19em, 6.153em, -999.998em); top: -4.471em; left: 3.264em;\"><span style=\"display: inline-block; position: relative; width: 2.192em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.264em, 1002.19em, 4.243em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-497\"><span class=\"mrow\" id=\"MathJax-Span-498\"><span class=\"texatom\" id=\"MathJax-Span-499\"><span class=\"mrow\" id=\"MathJax-Span-500\"><span class=\"mi\" id=\"MathJax-Span-501\" style=\"font-family: MathJax_Main;\">c<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-502\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"texatom\" id=\"MathJax-Span-503\" style=\"padding-left: 0.235em;\"><span class=\"mrow\" id=\"MathJax-Span-504\"><span class=\"mi\" id=\"MathJax-Span-505\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.264em, 1002.19em, 4.243em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-524\"><span class=\"mrow\" id=\"MathJax-Span-525\"><span class=\"texatom\" id=\"MathJax-Span-526\"><span class=\"mrow\" id=\"MathJax-Span-527\"><span class=\"mi\" id=\"MathJax-Span-528\" style=\"font-family: MathJax_Main;\">c<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-529\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"texatom\" id=\"MathJax-Span-530\" style=\"padding-left: 0.235em;\"><span class=\"mrow\" id=\"MathJax-Span-531\"><span class=\"mi\" id=\"MathJax-Span-532\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.264em, 1002.15em, 4.243em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-551\"><span class=\"mrow\" id=\"MathJax-Span-552\"><span class=\"texatom\" id=\"MathJax-Span-553\"><span class=\"mrow\" id=\"MathJax-Span-554\"><span class=\"mi\" id=\"MathJax-Span-555\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-556\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"texatom\" id=\"MathJax-Span-557\" style=\"padding-left: 0.235em;\"><span class=\"mrow\" id=\"MathJax-Span-558\"><span class=\"mi\" id=\"MathJax-Span-559\" style=\"font-family: MathJax_Main;\">c<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.379em, 1002.29em, 6.247em, -999.998em); top: -4.564em; left: 6.433em;\"><span style=\"display: inline-block; position: relative; width: 2.286em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1002.29em, 4.243em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-506\"><span class=\"mrow\" id=\"MathJax-Span-507\"><span class=\"texatom\" id=\"MathJax-Span-508\"><span class=\"mrow\" id=\"MathJax-Span-509\"><span class=\"mi\" id=\"MathJax-Span-510\" style=\"font-family: MathJax_Main;\">b<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-511\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"texatom\" id=\"MathJax-Span-512\" style=\"padding-left: 0.235em;\"><span class=\"mrow\" id=\"MathJax-Span-513\"><span class=\"mi\" id=\"MathJax-Span-514\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1002.29em, 4.243em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-533\"><span class=\"mrow\" id=\"MathJax-Span-534\"><span class=\"texatom\" id=\"MathJax-Span-535\"><span class=\"mrow\" id=\"MathJax-Span-536\"><span class=\"mi\" id=\"MathJax-Span-537\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-538\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"texatom\" id=\"MathJax-Span-539\" style=\"padding-left: 0.235em;\"><span class=\"mrow\" id=\"MathJax-Span-540\"><span class=\"mi\" id=\"MathJax-Span-541\" style=\"font-family: MathJax_Main;\">b<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1002.29em, 4.243em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-560\"><span class=\"mrow\" id=\"MathJax-Span-561\"><span class=\"texatom\" id=\"MathJax-Span-562\"><span class=\"mrow\" id=\"MathJax-Span-563\"><span class=\"mi\" id=\"MathJax-Span-564\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-565\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"texatom\" id=\"MathJax-Span-566\" style=\"padding-left: 0.235em;\"><span class=\"mrow\" id=\"MathJax-Span-567\"><span class=\"mi\" id=\"MathJax-Span-568\" style=\"font-family: MathJax_Main;\">b<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-569\" 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\">\\frac{1}{2}\\left[\\begin{array}{lll} \\mathrm{b}+\\mathrm{c} & \\mathrm{c}-\\mathrm{a} & \\mathrm{b}-\\mathrm{a} \\\\ \\mathrm{c}-\\mathrm{b} & \\mathrm{c}+\\mathrm{a} & \\mathrm{a}-\\mathrm{b} \\\\ \\mathrm{b}-\\mathrm{c} & \\mathrm{a}-\\mathrm{c} & \\mathrm{a}+\\mathrm{b} \\end{array}\\right]<\/script>, then show that ABA<sup>-1<\/sup> is a diagonal matrix.<\/h3>\n\n<p>Answer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-6216\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3e-8.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(e) 8\" width=\"335\" height=\"1122\" sizes=\"auto, (max-width: 335px) 100vw, 335px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(e) 8\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(e)\"><\/p>\n\n<h3>Question 2.<br> If 3A = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-15-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-570\" 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-571\"><span class=\"mrow\" id=\"MathJax-Span-572\"><span class=\"mo\" id=\"MathJax-Span-573\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Size4; top: -2.84em; left: 0em;\">\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-574\"><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, 1001.21em, 6.247em, -999.998em); top: -4.564em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-575\"><span class=\"mrow\" id=\"MathJax-Span-576\"><span class=\"mn\" id=\"MathJax-Span-577\" 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-584\"><span class=\"mrow\" id=\"MathJax-Span-585\"><span class=\"mn\" id=\"MathJax-Span-586\" 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-594\"><span class=\"mrow\" id=\"MathJax-Span-595\"><span class=\"mo\" id=\"MathJax-Span-596\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-597\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.06em, -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, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-578\"><span class=\"mrow\" id=\"MathJax-Span-579\"><span class=\"mn\" id=\"MathJax-Span-580\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-587\"><span class=\"mrow\" id=\"MathJax-Span-588\"><span class=\"mn\" id=\"MathJax-Span-589\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-598\"><span class=\"mrow\" id=\"MathJax-Span-599\"><span class=\"mn\" id=\"MathJax-Span-600\" 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(2.425em, 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-581\"><span class=\"mrow\" id=\"MathJax-Span-582\"><span class=\"mn\" id=\"MathJax-Span-583\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-590\"><span class=\"mrow\" id=\"MathJax-Span-591\"><span class=\"mo\" id=\"MathJax-Span-592\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-593\" 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-601\"><span class=\"mrow\" id=\"MathJax-Span-602\"><span class=\"mo\" id=\"MathJax-Span-603\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-604\" 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><\/span><span class=\"mo\" id=\"MathJax-Span-605\" 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}{ccc} 1 & 2 & 2 \\\\ 2 & 1 & -2 \\\\ -2 & 2 & -1 \\end{array}\\right]<\/script>, then show that A<sup>-I<\/sup> = A\u2019.<\/h3>\n\n<p>Answer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-6215\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3e-9.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(e) 9\" width=\"320\" height=\"343\" sizes=\"auto, (max-width: 320px) 100vw, 320px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(e) 9\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(e)\"><br> \u2234 A.A\u2019 = I and by definition A\u2019 = A<sup>-1<\/sup>\n\n<br> similarly A\u2019.A = I\n\n<\/p>\n\n<h3>Question 3.<br> If A = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-16-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-606\" 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-607\"><span class=\"mrow\" id=\"MathJax-Span-608\"><span class=\"mo\" id=\"MathJax-Span-609\" 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-610\"><span style=\"display: inline-block; position: relative; width: 4.289em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.107em, -999.998em); top: -4.471em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-611\"><span class=\"mrow\" id=\"MathJax-Span-612\"><span class=\"mn\" id=\"MathJax-Span-613\" style=\"font-family: MathJax_Main;\">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; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-621\"><span class=\"mrow\" id=\"MathJax-Span-622\"><span class=\"mn\" id=\"MathJax-Span-623\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-631\"><span class=\"mrow\" id=\"MathJax-Span-632\"><span class=\"mn\" id=\"MathJax-Span-633\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.425em, 1001.26em, 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, 1001.26em, 4.243em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-614\"><span class=\"mrow\" id=\"MathJax-Span-615\"><span class=\"mo\" id=\"MathJax-Span-616\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-617\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.26em, 4.243em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-624\"><span class=\"mrow\" id=\"MathJax-Span-625\"><span class=\"mo\" id=\"MathJax-Span-626\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-627\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-634\"><span class=\"mrow\" id=\"MathJax-Span-635\"><span class=\"mo\" id=\"MathJax-Span-636\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-637\" 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: 3.777em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-618\"><span class=\"mrow\" id=\"MathJax-Span-619\"><span class=\"mn\" id=\"MathJax-Span-620\" 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; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-628\"><span class=\"mrow\" id=\"MathJax-Span-629\"><span class=\"mn\" id=\"MathJax-Span-630\" style=\"font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-638\"><span class=\"mrow\" id=\"MathJax-Span-639\"><span class=\"mn\" id=\"MathJax-Span-640\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-641\" 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}{rrr} 3 & -3 & 4 \\\\ 2 & -3 & 4 \\\\ 0 & -1 & 1 \\end{array}\\right]<\/script>, then show that A<sup>-1<\/sup> = A<sup>3<\/sup><\/h3>\n\n<p>Answer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-6214\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3e-10.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(e) 10\" width=\"324\" height=\"635\" sizes=\"auto, (max-width: 324px) 100vw, 324px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(e) 10\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(e)\">\n\n<br> So, the multiplicative inverse of A exists and it is A<sup>3<\/sup>.<br> \u2234 A<sup>-1<\/sup> = A<sup>3<\/sup><\/p><p>Question 4.<br> If AB = I or BA = I, then prove that A is invertible and B = A<sup>-1<\/sup>.<br> Answer:<br> Given AB = I<br> \u21d2 |AB| = |I|<br> \u21d2 |A| |B| = 1<br> \u21d2 |A| \u2260 0<br> \u2234 A is a non-singular matrix.<br> Also BA = I<br> \u21d2 |B| |A| = |I|<br> \u21d2 |A| |B| = 1<br> \u21d2 |A| *0<br> \u2234 A is a non-singular matrix.<br> \u21d2 A is invertible<br> \u21d2 A<sup>-1<\/sup> exists AB = I<br> \u21d2 A<sup>-1<\/sup> AB = A<sup>-1<\/sup>I<br> \u21d2 (A<sup>-1<\/sup> A) B = A<sup>-1<\/sup>I<br> \u21d2 IB = A<sup>-1<\/sup>I<br> \u21d2 B = A<sup>-1<\/sup>.<\/p><\/div>\n","protected":false},"excerpt":{"rendered":"<p>TS Inter 1st Year Maths 1A Matrices Solutions Exercise 3(e) I. Question 1. Find the adjoint and inverse of the following matrices. (March 2002) i) [24\u221236] Answer: If A = [acbd] then adj A = [d\u2212c\u2212ba] ii) [cos\u03b1sin\u03b1\u2212sin\u03b1cos\u03b1] Answer: iii) Find the adjoint and inverse of the matrix \u23a1\u23a3\u23a2123012201\u23a4\u23a6\u23a5. Answer: Find cofactors of elements in [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":2711,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[1],"tags":[],"class_list":{"0":"post-2707","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-other"},"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(e)<\/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-3e\/\" \/>\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(e)\" \/>\r\n<meta property=\"og:description\" content=\"TS Inter 1st Year Maths 1A Matrices Solutions Exercise 3(e) I. Question 1. Find the adjoint and inverse of the following matrices. (March 2002) i) [24\u221236] Answer: If A = [acbd] then adj A = [d\u2212c\u2212ba] ii) [cos\u03b1sin\u03b1\u2212sin\u03b1cos\u03b1] Answer: iii) Find the adjoint and inverse of the matrix \u23a1\u23a3\u23a2123012201\u23a4\u23a6\u23a5. Answer: Find cofactors of elements in [&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-3e\/\" \/>\r\n<meta property=\"og:site_name\" content=\"Manabadi Boards\" \/>\r\n<meta property=\"article:published_time\" content=\"2026-01-10T07:15:07+00:00\" \/>\r\n<meta property=\"article:modified_time\" content=\"2026-01-10T07:15:09+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-3e.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 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Question 1. Find the adjoint and inverse of the following matrices. (March 2002) i) [24\u221236] Answer: If A = [acbd] then adj A = [d\u2212c\u2212ba] ii) [cos\u03b1sin\u03b1\u2212sin\u03b1cos\u03b1] Answer: iii) Find the adjoint and inverse of the matrix \u23a1\u23a3\u23a2123012201\u23a4\u23a6\u23a5. Answer: Find cofactors of elements in [&hellip;]","og_url":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3e\/","og_site_name":"Manabadi Boards","article_published_time":"2026-01-10T07:15:07+00:00","article_modified_time":"2026-01-10T07:15:09+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-3e.jpg","type":"image\/jpeg"}],"author":"Junaid","twitter_card":"summary_large_image","twitter_misc":{"Written by":"Junaid","Est. reading time":"5 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