{"id":2656,"date":"2026-01-09T14:57:40","date_gmt":"2026-01-09T09:27:40","guid":{"rendered":"https:\/\/www.manabadi.co.in\/boards\/?p=2656"},"modified":"2026-01-09T14:57:44","modified_gmt":"2026-01-09T09:27:44","slug":"ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3b","status":"publish","type":"post","link":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3b\/","title":{"rendered":"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b)"},"content":{"rendered":"\n \n\n<h2>TS Inter 1st Year Maths 1A Matrices Solutions Exercise 3(b)<\/h2>\n\n<h3>I.<br> Question 1.<br> Find the following products wherever possible.<br> i) [-1 4 2]<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: 2.565em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 2.192em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1001.87em, 6.2em, -999.998em); top: -4.425em; 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: 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-5\"><span style=\"display: inline-block; position: relative; width: 0.515em; 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; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-6\"><span class=\"mrow\" id=\"MathJax-Span-7\"><span class=\"mn\" id=\"MathJax-Span-8\" style=\"font-family: MathJax_Main;\">5<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-9\"><span class=\"mrow\" id=\"MathJax-Span-10\"><span class=\"mn\" id=\"MathJax-Span-11\" style=\"font-family: MathJax_Main;\">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-12\"><span class=\"mrow\" id=\"MathJax-Span-13\"><span class=\"mn\" id=\"MathJax-Span-14\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-15\" 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-1\">\\left[\\begin{array}{l}  5 \\\\  1 \\\\  3  \\end{array}\\right]<\/script><\/h3>\n\n<p>Answer:<br> It is a product of 1 \u00d7 3 and 3 \u00d7 1 matrices and the resulting is an 1 \u00d7 1 matrix.<br> [-1 4 2]<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-2-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-16\" style=\"width: 2.565em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 2.192em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1001.87em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-17\"><span class=\"mrow\" id=\"MathJax-Span-18\"><span class=\"mo\" id=\"MathJax-Span-19\" 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-20\"><span style=\"display: inline-block; position: relative; width: 0.515em; 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; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-21\"><span class=\"mrow\" id=\"MathJax-Span-22\"><span class=\"mn\" id=\"MathJax-Span-23\" style=\"font-family: MathJax_Main;\">5<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-24\"><span class=\"mrow\" id=\"MathJax-Span-25\"><span class=\"mn\" id=\"MathJax-Span-26\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-27\"><span class=\"mrow\" id=\"MathJax-Span-28\"><span class=\"mn\" id=\"MathJax-Span-29\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-30\" 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-2\">\\left[\\begin{array}{l}  5 \\\\  1 \\\\  3  \\end{array}\\right]<\/script><br> = [-1 \u00d7 5 + 4 \u00d7 1 + 2 \u00d7 3]<sub>1 \u00d7 1<\/sub> = [5]<sub>1 \u00d7 1<\/sub><\/p>\n\n<p>ii) <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-3-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-31\" style=\"width: 9.415em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 8.11em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.845em, 1007.78em, 6.899em, -999.998em); top: -5.123em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-32\"><span class=\"mrow\" id=\"MathJax-Span-33\"><span class=\"mo\" id=\"MathJax-Span-34\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-35\"><span style=\"display: inline-block; position: relative; width: 4.289em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.565em, 1000.47em, 4.942em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-36\"><span class=\"mrow\" id=\"MathJax-Span-37\"><span class=\"mn\" id=\"MathJax-Span-38\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-45\"><span class=\"mrow\" id=\"MathJax-Span-46\"><span class=\"mn\" id=\"MathJax-Span-47\" style=\"font-family: MathJax_Main;\">6<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1001.21em, 4.988em, -999.998em); top: -4.005em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-39\"><span class=\"mrow\" id=\"MathJax-Span-40\"><span class=\"mn\" id=\"MathJax-Span-41\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -3.26em; left: 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=\"mn\" id=\"MathJax-Span-51\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1000.47em, 4.942em, -999.998em); top: -4.005em; left: 3.777em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-42\"><span class=\"mrow\" id=\"MathJax-Span-43\"><span class=\"mn\" id=\"MathJax-Span-44\" style=\"font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-52\"><span class=\"mrow\" id=\"MathJax-Span-53\"><span class=\"mn\" id=\"MathJax-Span-54\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-55\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">]<\/span><\/span><\/span><span class=\"mrow\" id=\"MathJax-Span-56\" style=\"padding-left: 0.189em;\"><span class=\"mo\" id=\"MathJax-Span-57\" 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-58\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.06em, -999.998em); top: -4.471em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-59\"><span class=\"mrow\" id=\"MathJax-Span-60\"><span class=\"mn\" id=\"MathJax-Span-61\" 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-62\"><span class=\"mrow\" id=\"MathJax-Span-63\"><span class=\"mn\" id=\"MathJax-Span-64\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-65\"><span class=\"mrow\" id=\"MathJax-Span-66\"><span class=\"mn\" id=\"MathJax-Span-67\" 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-68\" 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: 5.128em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-3\">\\left[\\begin{array}{lll}  2 & 1 & 4 \\\\  6 & -2 & 3  \\end{array}\\right]\\left[\\begin{array}{l}  1 \\\\  2 \\\\  1  \\end{array}\\right]<\/script><br>\n\nAnswer:<br> It is a product of 2 \u00d7 3 and 3 \u00d7 1 matrices and the resulting is an 2 \u00d7 1 matrix.<br> <img decoding=\"async\" class=\"alignnone size-full wp-image-5741\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3b-1.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b) 1\" width=\"306\" height=\"144\" sizes=\"(max-width: 306px) 100vw, 306px\" data-pin-nopin=\"true\"><\/p>\n\n<p>iii) <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-4-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-69\" style=\"width: 10.068em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 8.67em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.773em, 1008.44em, 4.429em, -999.998em); top: -3.353em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-70\"><span class=\"mrow\" id=\"MathJax-Span-71\"><span class=\"mo\" id=\"MathJax-Span-72\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-73\"><span style=\"display: inline-block; position: relative; width: 2.798em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.565em, 1000.47em, 4.895em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -4.658em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-74\"><span class=\"mrow\" id=\"MathJax-Span-75\"><span class=\"mn\" id=\"MathJax-Span-76\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.26em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-81\"><span class=\"mrow\" id=\"MathJax-Span-82\"><span class=\"mn\" id=\"MathJax-Span-83\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1001.21em, 4.942em, -999.998em); top: -4.005em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -4.658em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-77\"><span class=\"mrow\" id=\"MathJax-Span-78\"><span class=\"mo\" id=\"MathJax-Span-79\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-80\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.26em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-84\"><span class=\"mrow\" id=\"MathJax-Span-85\"><span class=\"mn\" id=\"MathJax-Span-86\" style=\"font-family: MathJax_Main;\">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-87\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">]<\/span><\/span><\/span><span class=\"mrow\" id=\"MathJax-Span-88\" style=\"padding-left: 0.189em;\"><span class=\"mo\" id=\"MathJax-Span-89\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-90\"><span style=\"display: inline-block; position: relative; width: 2.798em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.565em, 1000.47em, 4.895em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -4.658em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-91\"><span class=\"mrow\" id=\"MathJax-Span-92\"><span class=\"mn\" id=\"MathJax-Span-93\" style=\"font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.26em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-98\"><span class=\"mrow\" id=\"MathJax-Span-99\"><span class=\"mn\" id=\"MathJax-Span-100\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1001.21em, 4.942em, -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, 1001.21em, 4.243em, -999.998em); top: -4.658em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-94\"><span class=\"mrow\" id=\"MathJax-Span-95\"><span class=\"mo\" id=\"MathJax-Span-96\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-97\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.26em; left: 50%; margin-left: -0.231em;\"><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;\">5<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-104\" 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}  3 & -2 \\\\  1 & 6  \\end{array}\\right]\\left[\\begin{array}{cc}  4 & -1 \\\\  2 & 5  \\end{array}\\right]<\/script><br>\n\nAnswer:<br> Product of 2 \u00d7 2 and 2 \u00d7 2 matrices and the resulting is an 2 \u00d7 2 matrix.<br> <img fetchpriority=\"high\" decoding=\"async\" class=\"alignnone size-full wp-image-5740\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3b-2.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b) 2\" width=\"270\" height=\"237\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b) 2\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b)\"><\/p>\n\n<p>iv) <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-5-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-105\" style=\"width: 15.007em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 12.91em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1012.58em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-106\"><span class=\"mrow\" id=\"MathJax-Span-107\"><span class=\"mo\" id=\"MathJax-Span-108\" 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-109\"><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-110\"><span class=\"mrow\" id=\"MathJax-Span-111\"><span class=\"mn\" id=\"MathJax-Span-112\" 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-119\"><span class=\"mrow\" id=\"MathJax-Span-120\"><span class=\"mn\" id=\"MathJax-Span-121\" 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-128\"><span class=\"mrow\" id=\"MathJax-Span-129\"><span class=\"mn\" id=\"MathJax-Span-130\" 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, 1000.47em, 4.15em, -999.998em); top: -5.356em; 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;\">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-122\"><span class=\"mrow\" id=\"MathJax-Span-123\"><span class=\"mn\" id=\"MathJax-Span-124\" 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-131\"><span class=\"mrow\" id=\"MathJax-Span-132\"><span class=\"mn\" id=\"MathJax-Span-133\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><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, 1000.42em, 4.15em, -999.998em); top: -5.356em; 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;\">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-125\"><span class=\"mrow\" id=\"MathJax-Span-126\"><span class=\"mn\" id=\"MathJax-Span-127\" 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-134\"><span class=\"mrow\" id=\"MathJax-Span-135\"><span class=\"mn\" id=\"MathJax-Span-136\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-137\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Size4; top: -2.84em; left: 0em;\">\u23a4<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Size4; top: -0.836em; left: 0em;\">\u23a6<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Size4; position: absolute; top: -1.815em; left: 0em;\">\u23a5<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span class=\"mrow\" id=\"MathJax-Span-138\" style=\"padding-left: 0.189em;\"><span class=\"mo\" id=\"MathJax-Span-139\" 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-140\"><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; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-141\"><span class=\"mrow\" id=\"MathJax-Span-142\"><span class=\"mo\" id=\"MathJax-Span-143\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-144\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-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 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-162\"><span class=\"mrow\" id=\"MathJax-Span-163\"><span class=\"mn\" id=\"MathJax-Span-164\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1001.26em, 6.06em, -999.998em); top: -4.471em; left: 2.286em;\"><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; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-145\"><span class=\"mrow\" id=\"MathJax-Span-146\"><span class=\"mo\" id=\"MathJax-Span-147\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-148\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-155\"><span class=\"mrow\" id=\"MathJax-Span-156\"><span class=\"mn\" id=\"MathJax-Span-157\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-165\"><span class=\"mrow\" id=\"MathJax-Span-166\"><span class=\"mn\" id=\"MathJax-Span-167\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><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: 4.569em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><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;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.26em, 4.243em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-158\"><span class=\"mrow\" id=\"MathJax-Span-159\"><span class=\"mo\" id=\"MathJax-Span-160\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-161\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-168\"><span class=\"mrow\" id=\"MathJax-Span-169\"><span class=\"mo\" id=\"MathJax-Span-170\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-171\" 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><\/span><span class=\"mo\" id=\"MathJax-Span-172\" 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}  2 & 2 & 1 \\\\  1 & 0 & 2 \\\\  2 & 1 & 2  \\end{array}\\right]\\left[\\begin{array}{ccc}  -2 & -3 & 4 \\\\  2 & 2 & -3 \\\\  1 & 2 & -2  \\end{array}\\right]<\/script><br>\n\nAnswer:<br> Product of 3 \u00d7 3 and 3 \u00d7 3 matrices and the resulting is an 3 \u00d7 3 matrix.<br> <img decoding=\"async\" class=\"alignnone size-full wp-image-5739\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3b-3.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b) 3\" width=\"265\" height=\"262\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b) 3\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b)\"><\/p>\n\n<p>v) <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-6-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-173\" style=\"width: 15.007em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 12.91em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.845em, 1012.68em, 6.899em, -999.998em); top: -5.123em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-174\"><span class=\"mrow\" id=\"MathJax-Span-175\"><span class=\"mo\" id=\"MathJax-Span-176\" 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-177\"><span style=\"display: inline-block; position: relative; width: 4.802em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.06em, -999.998em); top: -4.471em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-178\"><span class=\"mrow\" id=\"MathJax-Span-179\"><span class=\"mn\" id=\"MathJax-Span-180\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-187\"><span class=\"mrow\" id=\"MathJax-Span-188\"><span class=\"mn\" id=\"MathJax-Span-189\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-197\"><span class=\"mrow\" id=\"MathJax-Span-198\"><span class=\"mn\" id=\"MathJax-Span-199\" 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(2.379em, 1001.21em, 6.153em, -999.998em); top: -4.518em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-181\"><span class=\"mrow\" id=\"MathJax-Span-182\"><span class=\"mn\" id=\"MathJax-Span-183\" style=\"font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-190\"><span class=\"mrow\" id=\"MathJax-Span-191\"><span class=\"mo\" id=\"MathJax-Span-192\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-193\" 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-200\"><span class=\"mrow\" id=\"MathJax-Span-201\"><span class=\"mn\" id=\"MathJax-Span-202\" 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: 4.522em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.93em, 6.06em, -999.998em); top: -4.471em; left: 3.777em;\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-184\"><span class=\"mrow\" id=\"MathJax-Span-185\"><span class=\"mn\" id=\"MathJax-Span-186\" style=\"font-family: MathJax_Main;\">9<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-194\"><span class=\"mrow\" id=\"MathJax-Span-195\"><span class=\"mn\" id=\"MathJax-Span-196\" style=\"font-family: MathJax_Main;\">5<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.93em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.51em;\"><span class=\"mtd\" id=\"MathJax-Span-203\"><span class=\"mrow\" id=\"MathJax-Span-204\"><span class=\"mn\" id=\"MathJax-Span-205\" style=\"font-family: MathJax_Main;\">12<\/span><\/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-206\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Size4; top: -2.84em; left: 0em;\">\u23a4<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Size4; top: -0.836em; left: 0em;\">\u23a6<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Size4; position: absolute; top: -1.815em; left: 0em;\">\u23a5<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span class=\"mrow\" id=\"MathJax-Span-207\" style=\"padding-left: 0.189em;\"><span class=\"mo\" id=\"MathJax-Span-208\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-209\"><span style=\"display: inline-block; position: relative; width: 4.802em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.565em, 1000.98em, 4.942em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.98em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-210\"><span class=\"mrow\" id=\"MathJax-Span-211\"><span class=\"mn\" id=\"MathJax-Span-212\" style=\"font-family: MathJax_Main;\">13<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-220\"><span class=\"mrow\" id=\"MathJax-Span-221\"><span class=\"mn\" id=\"MathJax-Span-222\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1001.21em, 4.895em, -999.998em); top: -4.005em; left: 2.006em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-213\"><span class=\"mrow\" id=\"MathJax-Span-214\"><span class=\"mo\" id=\"MathJax-Span-215\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-216\" 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; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-223\"><span class=\"mrow\" id=\"MathJax-Span-224\"><span class=\"mn\" id=\"MathJax-Span-225\" style=\"font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1000.47em, 4.895em, -999.998em); top: -4.005em; left: 4.289em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-217\"><span class=\"mrow\" id=\"MathJax-Span-218\"><span class=\"mn\" id=\"MathJax-Span-219\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-226\"><span class=\"mrow\" id=\"MathJax-Span-227\"><span class=\"mn\" id=\"MathJax-Span-228\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-229\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">]<\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 5.128em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-6\">\\left[\\begin{array}{ccc}  3 & 4 & 9 \\\\  0 & -1 & 5 \\\\  2 & 6 & 12  \\end{array}\\right]\\left[\\begin{array}{lll}  13 & -2 & 0 \\\\  0 & 4 & 1  \\end{array}\\right]<\/script><br>\n\nAnswer:<br> Product of A<sub>3 \u00d7 3<\/sub> and B<sub>2 \u00d7 3<\/sub> matrices. Matrix multiplication is not confirmable since the number of columns of A \u2260 number of rows of B.<\/p><p>vi) <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-7-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-230\" style=\"width: 10.301em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 8.856em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.845em, 1008.62em, 6.899em, -999.998em); top: -5.123em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-231\"><span class=\"mrow\" id=\"MathJax-Span-232\"><span class=\"mo\" id=\"MathJax-Span-233\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Size4; top: -2.84em; left: 0em;\">\u23a1<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Size4; top: -0.836em; left: 0em;\">\u23a3<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Size4; position: absolute; top: -1.815em; left: 0em;\">\u23a2<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-234\"><span style=\"display: inline-block; position: relative; width: 1.26em; 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, 1000.42em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-235\"><span class=\"mrow\" id=\"MathJax-Span-236\"><span class=\"mn\" id=\"MathJax-Span-237\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-238\"><span class=\"mrow\" id=\"MathJax-Span-239\"><span class=\"mo\" id=\"MathJax-Span-240\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-241\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-242\"><span class=\"mrow\" id=\"MathJax-Span-243\"><span class=\"mn\" id=\"MathJax-Span-244\" 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-245\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Size4; top: -2.84em; left: 0em;\">\u23a4<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Size4; top: -0.836em; left: 0em;\">\u23a6<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Size4; position: absolute; top: -1.815em; left: 0em;\">\u23a5<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span class=\"mrow\" id=\"MathJax-Span-246\" style=\"padding-left: 0.189em;\"><span class=\"mo\" id=\"MathJax-Span-247\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-248\"><span style=\"display: inline-block; position: relative; width: 4.289em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.565em, 1000.47em, 4.942em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-249\"><span class=\"mrow\" id=\"MathJax-Span-250\"><span class=\"mn\" id=\"MathJax-Span-251\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-258\"><span class=\"mrow\" id=\"MathJax-Span-259\"><span class=\"mn\" id=\"MathJax-Span-260\" style=\"font-family: MathJax_Main;\">6<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1001.21em, 4.988em, -999.998em); top: -4.005em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-252\"><span class=\"mrow\" id=\"MathJax-Span-253\"><span class=\"mn\" id=\"MathJax-Span-254\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-261\"><span class=\"mrow\" id=\"MathJax-Span-262\"><span class=\"mo\" id=\"MathJax-Span-263\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-264\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1000.47em, 4.942em, -999.998em); top: -4.005em; left: 3.777em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-255\"><span class=\"mrow\" id=\"MathJax-Span-256\"><span class=\"mn\" id=\"MathJax-Span-257\" style=\"font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-265\"><span class=\"mrow\" id=\"MathJax-Span-266\"><span class=\"mn\" id=\"MathJax-Span-267\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-268\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">]<\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 5.128em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-7\">\\left[\\begin{array}{c}  1 \\\\  -2 \\\\  1  \\end{array}\\right]\\left[\\begin{array}{lll}  2 & 1 & 4 \\\\  6 & -2 & 3  \\end{array}\\right]<\/script><br>\n\nAnswer:<br> Product of A<sub>3 \u00d7 1<\/sub> and B<sub>2 \u00d7 3<\/sub> matrices. Matrix multiplication is not confirmable since the number of coloumn of A \u2260 number of rows of B.<\/p><p>vii) <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-8-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-269\" style=\"width: 10.021em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 8.623em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.773em, 1008.39em, 4.429em, -999.998em); top: -3.353em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-270\"><span class=\"mrow\" id=\"MathJax-Span-271\"><span class=\"mo\" id=\"MathJax-Span-272\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-273\"><span style=\"display: inline-block; position: relative; width: 3.544em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.565em, 1001.21em, 4.988em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -4.658em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-274\"><span class=\"mrow\" id=\"MathJax-Span-275\"><span class=\"mn\" id=\"MathJax-Span-276\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -3.26em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-281\"><span class=\"mrow\" id=\"MathJax-Span-282\"><span class=\"mo\" id=\"MathJax-Span-283\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-284\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1001.21em, 4.895em, -999.998em); top: -4.005em; left: 2.286em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -4.658em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-277\"><span class=\"mrow\" id=\"MathJax-Span-278\"><span class=\"mo\" id=\"MathJax-Span-279\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-280\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.26em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-285\"><span class=\"mrow\" id=\"MathJax-Span-286\"><span class=\"mn\" id=\"MathJax-Span-287\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-288\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">]<\/span><\/span><\/span><span class=\"mrow\" id=\"MathJax-Span-289\" style=\"padding-left: 0.189em;\"><span class=\"mo\" id=\"MathJax-Span-290\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-291\"><span style=\"display: inline-block; position: relative; width: 2.006em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.565em, 1000.42em, 4.895em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-292\"><span class=\"mrow\" id=\"MathJax-Span-293\"><span class=\"mn\" id=\"MathJax-Span-294\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-298\"><span class=\"mrow\" id=\"MathJax-Span-299\"><span class=\"mn\" id=\"MathJax-Span-300\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1000.42em, 4.895em, -999.998em); top: -4.005em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-295\"><span class=\"mrow\" id=\"MathJax-Span-296\"><span class=\"mn\" id=\"MathJax-Span-297\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-301\"><span class=\"mrow\" id=\"MathJax-Span-302\"><span class=\"mn\" id=\"MathJax-Span-303\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-304\" 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}{rr}  1 & -1 \\\\  -1 & 1  \\end{array}\\right]\\left[\\begin{array}{ll}  1 & 1 \\\\  1 & 1  \\end{array}\\right]<\/script><br>\n\nAnswer:<br> Product of 2 \u00d7 2 and 2 \u00d7 2 matrices and the resulting is an 2 \u00d7 2 matrix.<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-5738\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3b-4.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b) 4\" width=\"194\" height=\"115\" data-pin-nopin=\"true\"><\/p>\n\n<p>viii) <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-9-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-305\" style=\"width: 16.498em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 14.215em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1013.89em, 6.293em, -999.998em); top: -4.471em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-306\"><span class=\"mrow\" id=\"MathJax-Span-307\"><span class=\"mo\" id=\"MathJax-Span-308\" 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-309\"><span style=\"display: inline-block; position: relative; width: 5.734em; 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: 1.214em; 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-310\"><span class=\"mrow\" id=\"MathJax-Span-311\"><span class=\"mn\" id=\"MathJax-Span-312\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.264em, 1001.21em, 4.243em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.603em;\"><span class=\"mtd\" id=\"MathJax-Span-320\"><span class=\"mrow\" id=\"MathJax-Span-321\"><span class=\"mo\" id=\"MathJax-Span-322\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mi\" id=\"MathJax-Span-323\" style=\"font-family: MathJax_Math-italic;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.42em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-330\"><span class=\"mrow\" id=\"MathJax-Span-331\"><span class=\"mi\" id=\"MathJax-Span-332\" 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: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.425em, 1001.31em, 6.014em, -999.998em); top: -4.331em; left: 2.192em;\"><span style=\"display: inline-block; position: relative; width: 1.307em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.42em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-313\"><span class=\"mrow\" id=\"MathJax-Span-314\"><span class=\"mi\" id=\"MathJax-Span-315\" style=\"font-family: MathJax_Math-italic;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-324\"><span class=\"mrow\" id=\"MathJax-Span-325\"><span class=\"mn\" id=\"MathJax-Span-326\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.264em, 1001.31em, 4.243em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-333\"><span class=\"mrow\" id=\"MathJax-Span-334\"><span class=\"mo\" id=\"MathJax-Span-335\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mi\" id=\"MathJax-Span-336\" style=\"font-family: MathJax_Math-italic;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.336em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1001.21em, 6.153em, -999.998em); top: -4.518em; left: 4.522em;\"><span style=\"display: inline-block; position: relative; width: 1.214em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.21em, 4.243em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.603em;\"><span class=\"mtd\" id=\"MathJax-Span-316\"><span class=\"mrow\" id=\"MathJax-Span-317\"><span class=\"mo\" id=\"MathJax-Span-318\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mi\" id=\"MathJax-Span-319\" style=\"font-family: MathJax_Math-italic;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.277em;\"><span class=\"mtd\" id=\"MathJax-Span-327\"><span class=\"mrow\" id=\"MathJax-Span-328\"><span class=\"mi\" id=\"MathJax-Span-329\" style=\"font-family: MathJax_Math-italic;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.231em;\"><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;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.522em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-340\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Size4; top: -2.84em; left: 0em;\">\u23a4<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Size4; top: -0.836em; left: 0em;\">\u23a6<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Size4; position: absolute; top: -1.815em; left: 0em;\">\u23a5<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><span class=\"mrow\" id=\"MathJax-Span-341\" style=\"padding-left: 0.189em;\"><span class=\"mo\" id=\"MathJax-Span-342\" style=\"vertical-align: 2.192em;\"><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: -0.743em; left: 0em;\">\u23a3<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Size4; position: absolute; top: -1.768em; left: 0em;\">\u23a2<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-343\"><span style=\"display: inline-block; position: relative; width: 4.895em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.379em, 1000.98em, 6.34em, -999.998em); top: -4.704em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.031em, 1000.98em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.464em;\"><span class=\"mtd\" id=\"MathJax-Span-344\"><span class=\"mrow\" id=\"MathJax-Span-345\"><span class=\"msubsup\" id=\"MathJax-Span-346\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-347\" style=\"font-family: MathJax_Math-italic;\">a<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-348\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.93em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.464em;\"><span class=\"mtd\" id=\"MathJax-Span-357\"><span class=\"mrow\" id=\"MathJax-Span-358\"><span class=\"mi\" id=\"MathJax-Span-359\" style=\"font-family: MathJax_Math-italic;\">a<\/span><span class=\"mi\" id=\"MathJax-Span-360\" style=\"font-family: MathJax_Math-italic;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.98em, 4.15em, -999.998em); top: -2.514em; left: 50%; margin-left: -0.464em;\"><span class=\"mtd\" id=\"MathJax-Span-370\"><span class=\"mrow\" id=\"MathJax-Span-371\"><span class=\"mi\" id=\"MathJax-Span-372\" style=\"font-family: MathJax_Math-italic;\">a<\/span><span class=\"mi\" id=\"MathJax-Span-373\" 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: 4.709em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.379em, 1000.93em, 6.2em, -999.998em); top: -4.564em; left: 1.959em;\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.93em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.464em;\"><span class=\"mtd\" id=\"MathJax-Span-349\"><span class=\"mrow\" id=\"MathJax-Span-350\"><span class=\"mi\" id=\"MathJax-Span-351\" style=\"font-family: MathJax_Math-italic;\">a<\/span><span class=\"mi\" id=\"MathJax-Span-352\" style=\"font-family: MathJax_Math-italic;\">b<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.031em, 1000.84em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.417em;\"><span class=\"mtd\" id=\"MathJax-Span-361\"><span class=\"mrow\" id=\"MathJax-Span-362\"><span class=\"msubsup\" id=\"MathJax-Span-363\"><span style=\"display: inline-block; position: relative; width: 0.841em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-364\" style=\"font-family: MathJax_Math-italic;\">b<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.422em;\"><span class=\"mn\" id=\"MathJax-Span-365\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.84em, 4.15em, -999.998em); top: -2.514em; left: 50%; margin-left: -0.417em;\"><span class=\"mtd\" id=\"MathJax-Span-374\"><span class=\"mrow\" id=\"MathJax-Span-375\"><span class=\"mi\" id=\"MathJax-Span-376\" style=\"font-family: MathJax_Math-italic;\">b<\/span><span class=\"mi\" id=\"MathJax-Span-377\" 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: 4.569em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.379em, 1000.98em, 5.967em, -999.998em); top: -4.331em; left: 3.917em;\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.98em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.464em;\"><span class=\"mtd\" id=\"MathJax-Span-353\"><span class=\"mrow\" id=\"MathJax-Span-354\"><span class=\"mi\" id=\"MathJax-Span-355\" style=\"font-family: MathJax_Math-italic;\">a<\/span><span class=\"mi\" id=\"MathJax-Span-356\" style=\"font-family: MathJax_Math-italic;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.84em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.417em;\"><span class=\"mtd\" id=\"MathJax-Span-366\"><span class=\"mrow\" id=\"MathJax-Span-367\"><span class=\"mi\" id=\"MathJax-Span-368\" style=\"font-family: MathJax_Math-italic;\">b<\/span><span class=\"mi\" id=\"MathJax-Span-369\" style=\"font-family: MathJax_Math-italic;\">c<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.031em, 1000.84em, 4.15em, -999.998em); top: -2.514em; left: 50%; margin-left: -0.417em;\"><span class=\"mtd\" id=\"MathJax-Span-378\"><span class=\"mrow\" id=\"MathJax-Span-379\"><span class=\"msubsup\" id=\"MathJax-Span-380\"><span style=\"display: inline-block; position: relative; width: 0.841em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.451em, 1000.42em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-381\" style=\"font-family: MathJax_Math-italic;\">c<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 0.422em;\"><span class=\"mn\" id=\"MathJax-Span-382\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.336em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-383\" style=\"vertical-align: 2.192em;\"><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: -0.743em; left: 0em;\">\u23a6<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Size4; position: absolute; top: -1.768em; left: 0em;\">\u23a5<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.997em; border-left: 0px solid; width: 0px; height: 4.651em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-9\">\\left[\\begin{array}{ccc}  0 & c & -b \\\\  -c & 0 & a \\\\  b & -a & 0  \\end{array}\\right]\\left[\\begin{array}{ccc}  a^2 & a b & a c \\\\  a b & b^2 & b c \\\\  a c & b c & c^2  \\end{array}\\right]<\/script><br>\n\nAnswer:<br> Product of 3 \u00d7 3 and 3 \u00d7 3 matrices and the resulting matrix is an 3 \u00d7 3 matrix.<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-5737\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3b-5.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b) 5\" width=\"361\" height=\"263\" sizes=\"auto, (max-width: 361px) 100vw, 361px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b) 5\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b)\"><\/p>\n\n<h3>Question 2.<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-384\" style=\"width: 7.598em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 6.526em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.773em, 1006.29em, 4.429em, -999.998em); top: -3.353em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-385\"><span class=\"mrow\" id=\"MathJax-Span-386\"><span class=\"mo\" id=\"MathJax-Span-387\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-388\"><span style=\"display: inline-block; position: relative; width: 5.082em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.565em, 1001.26em, 4.988em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -4.658em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-389\"><span class=\"mrow\" id=\"MathJax-Span-390\"><span class=\"mn\" id=\"MathJax-Span-391\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1001.26em, 4.243em, -999.998em); top: -3.26em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-399\"><span class=\"mrow\" id=\"MathJax-Span-400\"><span class=\"mo\" id=\"MathJax-Span-401\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-402\" style=\"font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1001.21em, 4.895em, -999.998em); top: -4.005em; left: 2.286em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -4.658em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-392\"><span class=\"mrow\" id=\"MathJax-Span-393\"><span class=\"mo\" id=\"MathJax-Span-394\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-395\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.26em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-403\"><span class=\"mrow\" id=\"MathJax-Span-404\"><span class=\"mn\" id=\"MathJax-Span-405\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1000.47em, 4.942em, -999.998em); top: -4.005em; 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: -4.658em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-396\"><span class=\"mrow\" id=\"MathJax-Span-397\"><span class=\"mn\" id=\"MathJax-Span-398\" style=\"font-family: MathJax_Main;\">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-406\"><span class=\"mrow\" id=\"MathJax-Span-407\"><span class=\"mn\" id=\"MathJax-Span-408\" style=\"font-family: MathJax_Main;\">5<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-409\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">]<\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 3.357em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.132em; border-left: 0px solid; width: 0px; height: 2.922em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-10\">\\left[\\begin{array}{rrr}  1 & -2 & 3 \\\\  -4 & 2 & 5  \\end{array}\\right]<\/script> and B = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-11-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-410\" style=\"width: 4.289em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.684em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1003.36em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-411\"><span class=\"mrow\" id=\"MathJax-Span-412\"><span class=\"mo\" id=\"MathJax-Span-413\" 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-414\"><span style=\"display: inline-block; position: relative; width: 2.006em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 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-415\"><span class=\"mrow\" id=\"MathJax-Span-416\"><span class=\"mn\" id=\"MathJax-Span-417\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-421\"><span class=\"mrow\" id=\"MathJax-Span-422\"><span class=\"mn\" id=\"MathJax-Span-423\" style=\"font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-427\"><span class=\"mrow\" id=\"MathJax-Span-428\"><span class=\"mn\" id=\"MathJax-Span-429\" 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, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-418\"><span class=\"mrow\" id=\"MathJax-Span-419\"><span class=\"mn\" id=\"MathJax-Span-420\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-424\"><span class=\"mrow\" id=\"MathJax-Span-425\"><span class=\"mn\" id=\"MathJax-Span-426\" style=\"font-family: MathJax_Main;\">5<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-430\"><span class=\"mrow\" id=\"MathJax-Span-431\"><span class=\"mn\" id=\"MathJax-Span-432\" style=\"font-family: MathJax_Main;\">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-433\" 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}{ll}  2 & 3 \\\\  4 & 5 \\\\  2 & 1  \\end{array}\\right]<\/script> do AB and BA exist ? If they exist find them. Do A and B commute with respect to<br> multiplication.<\/h3>\n\n<p>Answer:<br> Given A = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-12-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-434\" style=\"width: 7.598em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 6.526em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.773em, 1006.29em, 4.429em, -999.998em); top: -3.353em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-435\"><span class=\"mrow\" id=\"MathJax-Span-436\"><span class=\"mo\" id=\"MathJax-Span-437\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-438\"><span style=\"display: inline-block; position: relative; width: 5.082em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.565em, 1001.26em, 4.988em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -4.658em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-439\"><span class=\"mrow\" id=\"MathJax-Span-440\"><span class=\"mn\" id=\"MathJax-Span-441\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1001.26em, 4.243em, -999.998em); top: -3.26em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-449\"><span class=\"mrow\" id=\"MathJax-Span-450\"><span class=\"mo\" id=\"MathJax-Span-451\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-452\" style=\"font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1001.21em, 4.895em, -999.998em); top: -4.005em; left: 2.286em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -4.658em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-442\"><span class=\"mrow\" id=\"MathJax-Span-443\"><span class=\"mo\" id=\"MathJax-Span-444\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-445\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.26em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-453\"><span class=\"mrow\" id=\"MathJax-Span-454\"><span class=\"mn\" id=\"MathJax-Span-455\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1000.47em, 4.942em, -999.998em); top: -4.005em; 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: -4.658em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-446\"><span class=\"mrow\" id=\"MathJax-Span-447\"><span class=\"mn\" id=\"MathJax-Span-448\" style=\"font-family: MathJax_Main;\">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-456\"><span class=\"mrow\" id=\"MathJax-Span-457\"><span class=\"mn\" id=\"MathJax-Span-458\" style=\"font-family: MathJax_Main;\">5<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-459\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">]<\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 3.357em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.132em; border-left: 0px solid; width: 0px; height: 2.922em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-12\">\\left[\\begin{array}{rrr}  1 & -2 & 3 \\\\  -4 & 2 & 5  \\end{array}\\right]<\/script> and B = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-13-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-460\" style=\"width: 4.289em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.684em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1003.36em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-461\"><span class=\"mrow\" id=\"MathJax-Span-462\"><span class=\"mo\" id=\"MathJax-Span-463\" 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-464\"><span style=\"display: inline-block; position: relative; width: 2.006em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 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-465\"><span class=\"mrow\" id=\"MathJax-Span-466\"><span class=\"mn\" id=\"MathJax-Span-467\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-471\"><span class=\"mrow\" id=\"MathJax-Span-472\"><span class=\"mn\" id=\"MathJax-Span-473\" style=\"font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-477\"><span class=\"mrow\" id=\"MathJax-Span-478\"><span class=\"mn\" id=\"MathJax-Span-479\" 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, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-468\"><span class=\"mrow\" id=\"MathJax-Span-469\"><span class=\"mn\" id=\"MathJax-Span-470\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-474\"><span class=\"mrow\" id=\"MathJax-Span-475\"><span class=\"mn\" id=\"MathJax-Span-476\" style=\"font-family: MathJax_Main;\">5<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-480\"><span class=\"mrow\" id=\"MathJax-Span-481\"><span class=\"mn\" id=\"MathJax-Span-482\" 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-483\" 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}{ll}  2 & 3 \\\\  4 & 5 \\\\  2 & 1  \\end{array}\\right]<\/script><br>\n\nWe have the product of A<sub>2\u00d73<\/sub> and B<sub>3\u00d72<\/sub> matrices and resulting AB is a product matrix of order 2 \u00d7 2. Similarly the product of B<sub>3\u00d72<\/sub> and A<sub>2\u00d73<\/sub> matrices results a product matrix BA of order 3 \u00d7 3.<br>\n\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-5736\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3b-6.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b) 6\" width=\"253\" height=\"475\" sizes=\"auto, (max-width: 253px) 100vw, 253px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b) 6\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b)\">\n\n<br> Since AB \u2260 BA, we have A and B are not com\u00acmutative with respect to multiplication of matrices.<\/p>\n\n<h3>Question 3.<br> Find A<sup>2<\/sup> where A = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-14-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-484\" 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-485\"><span class=\"mrow\" id=\"MathJax-Span-486\"><span class=\"mo\" id=\"MathJax-Span-487\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-488\"><span style=\"display: inline-block; position: relative; width: 2.798em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.565em, 1001.21em, 4.988em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -4.658em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-489\"><span class=\"mrow\" id=\"MathJax-Span-490\"><span class=\"mn\" id=\"MathJax-Span-491\" style=\"font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -3.26em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-495\"><span class=\"mrow\" id=\"MathJax-Span-496\"><span class=\"mo\" id=\"MathJax-Span-497\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-498\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1000.47em, 4.895em, -999.998em); top: -4.005em; left: 2.286em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -4.658em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-492\"><span class=\"mrow\" id=\"MathJax-Span-493\"><span class=\"mn\" id=\"MathJax-Span-494\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.26em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-499\"><span class=\"mrow\" id=\"MathJax-Span-500\"><span class=\"mn\" id=\"MathJax-Span-501\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-502\" 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-14\">\\left[\\begin{array}{cc}  4 & 2 \\\\  -1 & 1  \\end{array}\\right]<\/script><\/h3>\n\n<p>Answer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-5735\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3b-7.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b) 7\" width=\"286\" height=\"114\" data-pin-nopin=\"true\"><\/p>\n\n<h3>Question 4.<br> If A = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-15-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-503\" style=\"width: 4.01em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.451em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.773em, 1003.22em, 4.429em, -999.998em); top: -3.353em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-504\"><span class=\"mrow\" id=\"MathJax-Span-505\"><span class=\"mo\" id=\"MathJax-Span-506\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-507\"><span style=\"display: inline-block; position: relative; width: 2.006em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.565em, 1000.47em, 4.942em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.24em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-508\"><span class=\"mrow\" id=\"MathJax-Span-509\"><span class=\"texatom\" id=\"MathJax-Span-510\"><span class=\"mrow\" id=\"MathJax-Span-511\"><span class=\"mi\" id=\"MathJax-Span-512\" style=\"font-family: MathJax_Main;\">i<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-516\"><span class=\"mrow\" id=\"MathJax-Span-517\"><span class=\"mn\" id=\"MathJax-Span-518\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1000.47em, 4.895em, -999.998em); top: -4.005em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-513\"><span class=\"mrow\" id=\"MathJax-Span-514\"><span class=\"mn\" id=\"MathJax-Span-515\" 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, 1000.24em, 4.15em, -999.998em); top: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-519\"><span class=\"mrow\" id=\"MathJax-Span-520\"><span class=\"texatom\" id=\"MathJax-Span-521\"><span class=\"mrow\" id=\"MathJax-Span-522\"><span class=\"mi\" id=\"MathJax-Span-523\" style=\"font-family: MathJax_Main;\">i<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-524\" 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-15\">\\left[\\begin{array}{ll}  \\mathrm{i} & 0 \\\\  0 & \\mathrm{i}  \\end{array}\\right]<\/script> find A<sup>2<\/sup>.<\/h3>\n\n<p>Answer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-5734\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3b-8.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b) 8\" width=\"204\" height=\"182\" data-pin-nopin=\"true\"><\/p>\n\n<h3>Question 5.<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-525\" style=\"width: 4.662em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.01em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.773em, 1003.78em, 4.429em, -999.998em); top: -3.353em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-526\"><span class=\"mrow\" id=\"MathJax-Span-527\"><span class=\"mo\" id=\"MathJax-Span-528\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-529\"><span style=\"display: inline-block; position: relative; width: 2.565em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.565em, 1000.47em, 4.942em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.24em, 4.15em, -999.998em); top: -4.658em; left: 50%; margin-left: -0.137em;\"><span class=\"mtd\" id=\"MathJax-Span-530\"><span class=\"mrow\" id=\"MathJax-Span-531\"><span class=\"texatom\" id=\"MathJax-Span-532\"><span class=\"mrow\" id=\"MathJax-Span-533\"><span class=\"mi\" id=\"MathJax-Span-534\" style=\"font-family: MathJax_Main;\">i<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.26em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-538\"><span class=\"mrow\" id=\"MathJax-Span-539\"><span class=\"mn\" id=\"MathJax-Span-540\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1001.03em, 4.988em, -999.998em); top: -4.005em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 1.074em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -4.658em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-535\"><span class=\"mrow\" id=\"MathJax-Span-536\"><span class=\"mn\" id=\"MathJax-Span-537\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.03em, 4.243em, -999.998em); top: -3.26em; left: 50%; margin-left: -0.51em;\"><span class=\"mtd\" id=\"MathJax-Span-541\"><span class=\"mrow\" id=\"MathJax-Span-542\"><span class=\"mo\" id=\"MathJax-Span-543\" style=\"font-family: MathJax_Main;\">\u2212<\/span><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;\">i<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-547\" 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-16\">\\left[\\begin{array}{cc}  \\mathrm{i} & 0 \\\\  0 & -\\mathrm{i}  \\end{array}\\right]<\/script>; B = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-17-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-548\" 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-549\"><span class=\"mrow\" id=\"MathJax-Span-550\"><span class=\"mo\" id=\"MathJax-Span-551\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-552\"><span style=\"display: inline-block; position: relative; width: 2.798em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.565em, 1000.47em, 4.895em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -4.658em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-553\"><span class=\"mrow\" id=\"MathJax-Span-554\"><span class=\"mn\" id=\"MathJax-Span-555\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.26em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-560\"><span class=\"mrow\" id=\"MathJax-Span-561\"><span class=\"mn\" id=\"MathJax-Span-562\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1001.21em, 4.942em, -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, 1001.21em, 4.243em, -999.998em); top: -4.658em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-556\"><span class=\"mrow\" id=\"MathJax-Span-557\"><span class=\"mo\" id=\"MathJax-Span-558\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-559\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.26em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-563\"><span class=\"mrow\" id=\"MathJax-Span-564\"><span class=\"mn\" id=\"MathJax-Span-565\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-566\" 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-17\">\\left[\\begin{array}{cc}  0 & -1 \\\\  1 & 0  \\end{array}\\right]<\/script> and C = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-18-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-567\" style=\"width: 4.01em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.451em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.773em, 1003.22em, 4.429em, -999.998em); top: -3.353em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-568\"><span class=\"mrow\" id=\"MathJax-Span-569\"><span class=\"mo\" id=\"MathJax-Span-570\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-571\"><span style=\"display: inline-block; position: relative; width: 2.006em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.565em, 1000.47em, 4.895em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-572\"><span class=\"mrow\" id=\"MathJax-Span-573\"><span class=\"mn\" id=\"MathJax-Span-574\" 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, 1000.24em, 4.15em, -999.998em); top: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-580\"><span class=\"mrow\" id=\"MathJax-Span-581\"><span class=\"texatom\" id=\"MathJax-Span-582\"><span class=\"mrow\" id=\"MathJax-Span-583\"><span class=\"mi\" id=\"MathJax-Span-584\" style=\"font-family: MathJax_Main;\">i<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1000.47em, 4.942em, -999.998em); top: -4.005em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.24em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-575\"><span class=\"mrow\" id=\"MathJax-Span-576\"><span class=\"texatom\" id=\"MathJax-Span-577\"><span class=\"mrow\" id=\"MathJax-Span-578\"><span class=\"mi\" id=\"MathJax-Span-579\" style=\"font-family: MathJax_Main;\">i<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-585\"><span class=\"mrow\" id=\"MathJax-Span-586\"><span class=\"mn\" id=\"MathJax-Span-587\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-588\" 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-18\">\\left[\\begin{array}{ll}  0 & \\mathrm{i} \\\\  \\mathrm{i} & 0  \\end{array}\\right]<\/script> and I is the unit matrix of order 2, then show that<br> i) A<sup>2<\/sup> = B<sup>2<\/sup> = C<sup>2<\/sup> = \u2013 I<\/h3>\n\n<p>Answer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-5733\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3b-9.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b) 9\" width=\"217\" height=\"372\" sizes=\"auto, (max-width: 217px) 100vw, 217px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b) 9\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b)\"><\/p>\n\n<p>ii) AB = \u2013 BA = \u2013 C (March 2008)<br>\n\nAnswer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-5732\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3b-10.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b) 10\" width=\"205\" height=\"387\" sizes=\"auto, (max-width: 205px) 100vw, 205px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b) 10\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b)\"><\/p>\n\n<h3>Question 6.<br> If A = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-19-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-589\" style=\"width: 4.01em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.451em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.773em, 1003.22em, 4.429em, -999.998em); top: -3.353em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-590\"><span class=\"mrow\" id=\"MathJax-Span-591\"><span class=\"mo\" id=\"MathJax-Span-592\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-593\"><span style=\"display: inline-block; position: relative; width: 2.006em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.565em, 1000.47em, 4.895em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-594\"><span class=\"mrow\" id=\"MathJax-Span-595\"><span class=\"mn\" id=\"MathJax-Span-596\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-600\"><span class=\"mrow\" id=\"MathJax-Span-601\"><span class=\"mn\" id=\"MathJax-Span-602\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1000.47em, 4.942em, -999.998em); top: -4.005em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-597\"><span class=\"mrow\" id=\"MathJax-Span-598\"><span class=\"mn\" id=\"MathJax-Span-599\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-603\"><span class=\"mrow\" id=\"MathJax-Span-604\"><span class=\"mn\" id=\"MathJax-Span-605\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-606\" 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-19\">\\left[\\begin{array}{ll}  2 & 1 \\\\  1 & 3  \\end{array}\\right]<\/script> and B = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-20-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-607\" style=\"width: 5.734em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.942em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.773em, 1004.71em, 4.429em, -999.998em); top: -3.353em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-608\"><span class=\"mrow\" id=\"MathJax-Span-609\"><span class=\"mo\" id=\"MathJax-Span-610\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-611\"><span style=\"display: inline-block; position: relative; width: 3.497em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.565em, 1000.47em, 4.895em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-612\"><span class=\"mrow\" id=\"MathJax-Span-613\"><span class=\"mn\" id=\"MathJax-Span-614\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.26em; left: 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;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1000.47em, 4.942em, -999.998em); top: -4.005em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-615\"><span class=\"mrow\" id=\"MathJax-Span-616\"><span class=\"mn\" id=\"MathJax-Span-617\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-624\"><span class=\"mrow\" id=\"MathJax-Span-625\"><span class=\"mn\" id=\"MathJax-Span-626\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1000.47em, 4.895em, -999.998em); top: -4.005em; left: 2.985em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-618\"><span class=\"mrow\" id=\"MathJax-Span-619\"><span class=\"mn\" id=\"MathJax-Span-620\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-627\"><span class=\"mrow\" id=\"MathJax-Span-628\"><span class=\"mn\" id=\"MathJax-Span-629\" style=\"font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-630\" 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-20\">\\left[\\begin{array}{lll}  3 & 2 & 0 \\\\  1 & 0 & 4  \\end{array}\\right]<\/script> find AB. Find BA if it exists.<br> Answer:<br> Given A = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-21-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-631\" style=\"width: 4.01em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.451em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.773em, 1003.22em, 4.429em, -999.998em); top: -3.353em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-632\"><span class=\"mrow\" id=\"MathJax-Span-633\"><span class=\"mo\" id=\"MathJax-Span-634\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-635\"><span style=\"display: inline-block; position: relative; width: 2.006em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.565em, 1000.47em, 4.895em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-636\"><span class=\"mrow\" id=\"MathJax-Span-637\"><span class=\"mn\" id=\"MathJax-Span-638\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-642\"><span class=\"mrow\" id=\"MathJax-Span-643\"><span class=\"mn\" id=\"MathJax-Span-644\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1000.47em, 4.942em, -999.998em); top: -4.005em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-639\"><span class=\"mrow\" id=\"MathJax-Span-640\"><span class=\"mn\" id=\"MathJax-Span-641\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-645\"><span class=\"mrow\" id=\"MathJax-Span-646\"><span class=\"mn\" id=\"MathJax-Span-647\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-648\" 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-21\">\\left[\\begin{array}{ll}  2 & 1 \\\\  1 & 3  \\end{array}\\right]<\/script> and B = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-22-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-649\" style=\"width: 5.734em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.942em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.773em, 1004.71em, 4.429em, -999.998em); top: -3.353em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-650\"><span class=\"mrow\" id=\"MathJax-Span-651\"><span class=\"mo\" id=\"MathJax-Span-652\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-653\"><span style=\"display: inline-block; position: relative; width: 3.497em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.565em, 1000.47em, 4.895em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-654\"><span class=\"mrow\" id=\"MathJax-Span-655\"><span class=\"mn\" id=\"MathJax-Span-656\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-663\"><span class=\"mrow\" id=\"MathJax-Span-664\"><span class=\"mn\" id=\"MathJax-Span-665\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1000.47em, 4.942em, -999.998em); top: -4.005em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-657\"><span class=\"mrow\" id=\"MathJax-Span-658\"><span class=\"mn\" id=\"MathJax-Span-659\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-666\"><span class=\"mrow\" id=\"MathJax-Span-667\"><span class=\"mn\" id=\"MathJax-Span-668\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1000.47em, 4.895em, -999.998em); top: -4.005em; left: 2.985em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-660\"><span class=\"mrow\" id=\"MathJax-Span-661\"><span class=\"mn\" id=\"MathJax-Span-662\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-669\"><span class=\"mrow\" id=\"MathJax-Span-670\"><span class=\"mn\" id=\"MathJax-Span-671\" style=\"font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-672\" 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-22\">\\left[\\begin{array}{lll}  3 & 2 & 0 \\\\  1 & 0 & 4  \\end{array}\\right]<\/script> are matrices of 2 \u00d7 2 and 2 \u00d7 3. The resulting matrix AB is of the form 2 \u00d7 3.<br>\n\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-5731\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3b-11.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b) 11\" width=\"199\" height=\"172\" data-pin-nopin=\"true\"><\/p>\n\n<h3>Question 7.<br> If A = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-23-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-673\" style=\"width: 5.361em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.616em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.773em, 1004.38em, 4.429em, -999.998em); top: -3.353em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-674\"><span class=\"mrow\" id=\"MathJax-Span-675\"><span class=\"mo\" id=\"MathJax-Span-676\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-677\"><span style=\"display: inline-block; position: relative; width: 3.171em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.565em, 1001.21em, 4.988em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -4.658em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-678\"><span class=\"mrow\" id=\"MathJax-Span-679\"><span class=\"mn\" id=\"MathJax-Span-680\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -3.26em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-684\"><span class=\"mrow\" id=\"MathJax-Span-685\"><span class=\"mo\" id=\"MathJax-Span-686\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-687\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1000.89em, 4.895em, -999.998em); top: -4.005em; left: 2.286em;\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -4.658em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-681\"><span class=\"mrow\" id=\"MathJax-Span-682\"><span class=\"mn\" id=\"MathJax-Span-683\" 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, 1000.89em, 4.15em, -999.998em); top: -3.26em; left: 50%; margin-left: -0.464em;\"><span class=\"mtd\" id=\"MathJax-Span-688\"><span class=\"mrow\" id=\"MathJax-Span-689\"><span class=\"mi\" id=\"MathJax-Span-690\" style=\"font-family: MathJax_Math-italic;\">K<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.049em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-691\" 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-23\">\\left[\\begin{array}{cc}  2 & 4 \\\\  -1 & K  \\end{array}\\right]<\/script> and A<sup>2<\/sup> = 0 then find the value of K. (May 2011, Mar. \u201914, \u201905)<\/h3>\n\n<p>Answer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-5730\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3b-12.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b) 12\" width=\"269\" height=\"158\" data-pin-nopin=\"true\"><\/p>\n\n<h3>II.<br> Question 1.<br> If A = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-24-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-692\" 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-693\"><span class=\"mrow\" id=\"MathJax-Span-694\"><span class=\"mo\" id=\"MathJax-Span-695\" 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-696\"><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, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-697\"><span class=\"mrow\" id=\"MathJax-Span-698\"><span class=\"mn\" id=\"MathJax-Span-699\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-706\"><span class=\"mrow\" id=\"MathJax-Span-707\"><span class=\"mn\" id=\"MathJax-Span-708\" 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-715\"><span class=\"mrow\" id=\"MathJax-Span-716\"><span class=\"mn\" id=\"MathJax-Span-717\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.107em, -999.998em); top: -4.471em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-700\"><span class=\"mrow\" id=\"MathJax-Span-701\"><span class=\"mn\" id=\"MathJax-Span-702\" 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-709\"><span class=\"mrow\" id=\"MathJax-Span-710\"><span class=\"mn\" id=\"MathJax-Span-711\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-718\"><span class=\"mrow\" id=\"MathJax-Span-719\"><span class=\"mn\" id=\"MathJax-Span-720\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.107em, -999.998em); top: -4.471em; left: 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-703\"><span class=\"mrow\" id=\"MathJax-Span-704\"><span class=\"mn\" id=\"MathJax-Span-705\" 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-712\"><span class=\"mrow\" id=\"MathJax-Span-713\"><span class=\"mn\" id=\"MathJax-Span-714\" 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-721\"><span class=\"mrow\" id=\"MathJax-Span-722\"><span class=\"mn\" id=\"MathJax-Span-723\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-724\" 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-24\">\\left[\\begin{array}{lll}  3 & 0 & 0 \\\\  0 & 3 & 0 \\\\  0 & 0 & 3  \\end{array}\\right]<\/script> there find A<sup>4<\/sup>.<\/h3>\n\n<p>Answer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-5729\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3b-13.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b) 13\" width=\"288\" height=\"293\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b) 13\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b)\"><\/p>\n\n<h3>Question 2.<br> If A = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-25-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-725\" 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-726\"><span class=\"mrow\" id=\"MathJax-Span-727\"><span class=\"mo\" id=\"MathJax-Span-728\" 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-729\"><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(2.425em, 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-730\"><span class=\"mrow\" id=\"MathJax-Span-731\"><span class=\"mn\" id=\"MathJax-Span-732\" 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-739\"><span class=\"mrow\" id=\"MathJax-Span-740\"><span class=\"mn\" id=\"MathJax-Span-741\" style=\"font-family: MathJax_Main;\">5<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-748\"><span class=\"mrow\" id=\"MathJax-Span-749\"><span class=\"mo\" id=\"MathJax-Span-750\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-751\" 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(2.425em, 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, 1000.42em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-733\"><span class=\"mrow\" id=\"MathJax-Span-734\"><span class=\"mn\" id=\"MathJax-Span-735\" 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-742\"><span class=\"mrow\" id=\"MathJax-Span-743\"><span class=\"mn\" id=\"MathJax-Span-744\" 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-752\"><span class=\"mrow\" id=\"MathJax-Span-753\"><span class=\"mo\" id=\"MathJax-Span-754\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-755\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.425em, 1001.26em, 6.247em, -999.998em); top: -4.564em; left: 4.569em;\"><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-736\"><span class=\"mrow\" id=\"MathJax-Span-737\"><span class=\"mn\" id=\"MathJax-Span-738\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-745\"><span class=\"mrow\" id=\"MathJax-Span-746\"><span class=\"mn\" id=\"MathJax-Span-747\" style=\"font-family: MathJax_Main;\">6<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.26em, 4.243em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-756\"><span class=\"mrow\" id=\"MathJax-Span-757\"><span class=\"mo\" id=\"MathJax-Span-758\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-759\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-760\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Size4; top: -2.84em; left: 0em;\">\u23a4<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Size4; top: -0.836em; left: 0em;\">\u23a6<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Size4; position: absolute; top: -1.815em; left: 0em;\">\u23a5<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-25\">\\left[\\begin{array}{ccc}  1 & 1 & 3 \\\\  5 & 2 & 6 \\\\  -2 & -1 & -3  \\end{array}\\right]<\/script> then Find A<sup>3<\/sup>.<\/h3>\n\n<p>Answer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-5728\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3b-14.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b) 14\" width=\"328\" height=\"607\"  sizes=\"auto, (max-width: 328px) 100vw, 328px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b) 14\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b)\"><\/p>\n\n<h3>Question 3.<br> If A = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-26-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-761\" 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-762\"><span class=\"mrow\" id=\"MathJax-Span-763\"><span class=\"mo\" id=\"MathJax-Span-764\" 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-765\"><span style=\"display: inline-block; position: relative; width: 5.082em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.107em, -999.998em); top: -4.471em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-766\"><span class=\"mrow\" id=\"MathJax-Span-767\"><span class=\"mn\" id=\"MathJax-Span-768\" 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-776\"><span class=\"mrow\" id=\"MathJax-Span-777\"><span class=\"mn\" id=\"MathJax-Span-778\" 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; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-786\"><span class=\"mrow\" id=\"MathJax-Span-787\"><span class=\"mn\" id=\"MathJax-Span-788\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.425em, 1001.21em, 6.247em, -999.998em); top: -4.564em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-769\"><span class=\"mrow\" id=\"MathJax-Span-770\"><span class=\"mo\" id=\"MathJax-Span-771\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-772\" 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-779\"><span class=\"mrow\" id=\"MathJax-Span-780\"><span class=\"mn\" id=\"MathJax-Span-781\" 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-789\"><span class=\"mrow\" id=\"MathJax-Span-790\"><span class=\"mo\" id=\"MathJax-Span-791\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-792\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1001.21em, 6.06em, -999.998em); top: -4.471em; left: 3.777em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-773\"><span class=\"mrow\" id=\"MathJax-Span-774\"><span class=\"mn\" id=\"MathJax-Span-775\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-782\"><span class=\"mrow\" id=\"MathJax-Span-783\"><span class=\"mo\" id=\"MathJax-Span-784\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-785\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-793\"><span class=\"mrow\" id=\"MathJax-Span-794\"><span class=\"mn\" id=\"MathJax-Span-795\" 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-796\" 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-26\">\\left[\\begin{array}{rrr}  1 & -2 & 1 \\\\  0 & 1 & -1 \\\\  3 & -1 & 1  \\end{array}\\right]<\/script> then find A<sup>3<\/sup> \u2013 3A<sup>2<\/sup> \u2013 A \u2013 3I, where I is a unit matrix of order 3. (March 2011)<\/h3>\n\n<p>Answer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-5727\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3b-15.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b) 15\" width=\"367\" height=\"1192\" sizes=\"auto, (max-width: 367px) 100vw, 367px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b) 15\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b)\"><\/p>\n\n<h3>Question 4.<br> If I = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-27-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-797\" style=\"width: 4.01em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.451em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.773em, 1003.22em, 4.429em, -999.998em); top: -3.353em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-798\"><span class=\"mrow\" id=\"MathJax-Span-799\"><span class=\"mo\" id=\"MathJax-Span-800\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-801\"><span style=\"display: inline-block; position: relative; width: 2.006em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.565em, 1000.47em, 4.942em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-802\"><span class=\"mrow\" id=\"MathJax-Span-803\"><span class=\"mn\" id=\"MathJax-Span-804\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-808\"><span class=\"mrow\" id=\"MathJax-Span-809\"><span class=\"mn\" id=\"MathJax-Span-810\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1000.47em, 4.895em, -999.998em); top: -4.005em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-805\"><span class=\"mrow\" id=\"MathJax-Span-806\"><span class=\"mn\" id=\"MathJax-Span-807\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-811\"><span class=\"mrow\" id=\"MathJax-Span-812\"><span class=\"mn\" id=\"MathJax-Span-813\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-814\" 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-27\">\\left[\\begin{array}{ll}  1 & 0 \\\\  0 & 1  \\end{array}\\right]<\/script> and E = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-28-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-815\" style=\"width: 4.01em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.451em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.773em, 1003.22em, 4.429em, -999.998em); top: -3.353em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-816\"><span class=\"mrow\" id=\"MathJax-Span-817\"><span class=\"mo\" id=\"MathJax-Span-818\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-819\"><span style=\"display: inline-block; position: relative; width: 2.006em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.565em, 1000.47em, 4.942em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-820\"><span class=\"mrow\" id=\"MathJax-Span-821\"><span class=\"mn\" id=\"MathJax-Span-822\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-826\"><span class=\"mrow\" id=\"MathJax-Span-827\"><span class=\"mn\" id=\"MathJax-Span-828\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1000.47em, 4.942em, -999.998em); top: -4.005em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-823\"><span class=\"mrow\" id=\"MathJax-Span-824\"><span class=\"mn\" id=\"MathJax-Span-825\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-829\"><span class=\"mrow\" id=\"MathJax-Span-830\"><span class=\"mn\" id=\"MathJax-Span-831\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-832\" 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-28\">\\left[\\begin{array}{ll}  0 & 1 \\\\  0 & 0  \\end{array}\\right]<\/script> show that (aI + bE) = a<sup>3<\/sup>I + 3a<sup>2<\/sup>bE, where I is a unit matrix of order 2. (Mar. 2015-A.P)(May \u201905)|<\/h3>\n\n<p>Answer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-5726\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3b-16.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b) 16\" width=\"297\" height=\"488\" sizes=\"auto, (max-width: 297px) 100vw, 297px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b) 16\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b)\"><\/p>\n\n<h3>III.<br> Question 1.<br> If A = <sub>diag<\/sub>[a<sub>1<\/sub> a<sub>2<\/sub> a<sub>3<\/sub>] then for any Integer n \u2265 1 show that A<sup>n<\/sup> = <sub>diag<\/sub><span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-29-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-833\" style=\"width: 5.454em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.709em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.447em, 1004.52em, 2.938em, -999.998em); top: -2.421em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-834\"><span class=\"mrow\" id=\"MathJax-Span-835\"><span class=\"mo\" id=\"MathJax-Span-836\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size1;\">[<\/span><\/span><span class=\"msubsup\" id=\"MathJax-Span-837\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-838\"><span class=\"mrow\" id=\"MathJax-Span-839\"><span class=\"mi\" id=\"MathJax-Span-840\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.544em, 1000.47em, 4.15em, -999.998em); top: -4.331em; left: 0.515em;\"><span class=\"texatom\" id=\"MathJax-Span-841\"><span class=\"mrow\" id=\"MathJax-Span-842\"><span class=\"texatom\" id=\"MathJax-Span-843\"><span class=\"mrow\" id=\"MathJax-Span-844\"><span class=\"mi\" id=\"MathJax-Span-845\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">n<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.42em, 4.15em, -999.998em); top: -3.679em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-846\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-847\" style=\"font-family: MathJax_Main;\">,<\/span><span class=\"msubsup\" id=\"MathJax-Span-848\" style=\"padding-left: 0.189em;\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-849\"><span class=\"mrow\" id=\"MathJax-Span-850\"><span class=\"mi\" id=\"MathJax-Span-851\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.544em, 1000.47em, 4.15em, -999.998em); top: -4.331em; left: 0.515em;\"><span class=\"texatom\" id=\"MathJax-Span-852\"><span class=\"mrow\" id=\"MathJax-Span-853\"><span class=\"texatom\" id=\"MathJax-Span-854\"><span class=\"mrow\" id=\"MathJax-Span-855\"><span class=\"mi\" id=\"MathJax-Span-856\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">n<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.42em, 4.15em, -999.998em); top: -3.679em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-857\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-858\" style=\"font-family: MathJax_Main;\">,<\/span><span class=\"msubsup\" id=\"MathJax-Span-859\" style=\"padding-left: 0.189em;\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-860\"><span class=\"mrow\" id=\"MathJax-Span-861\"><span class=\"mi\" id=\"MathJax-Span-862\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.544em, 1000.47em, 4.15em, -999.998em); top: -4.331em; left: 0.515em;\"><span class=\"texatom\" id=\"MathJax-Span-863\"><span class=\"mrow\" id=\"MathJax-Span-864\"><span class=\"texatom\" id=\"MathJax-Span-865\"><span class=\"mrow\" id=\"MathJax-Span-866\"><span class=\"mi\" id=\"MathJax-Span-867\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">n<\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.42em, 4.15em, -999.998em); top: -3.679em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-868\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-869\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size1;\">]<\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.425em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.516em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-29\">\\left[\\mathrm{a}_1^{\\mathrm{n}}, \\mathrm{a}_2^{\\mathrm{n}}, \\mathrm{a}_3^{\\mathrm{n}}\\right]<\/script><\/h3>\n\n<p>Answer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-5725\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3b-17.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b) 17\" width=\"299\" height=\"216\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b) 17\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b)\"><br>\n\nWe prove this result by using mathematical induction suppose n = 1 then<br> A\u2019 = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-30-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-870\" style=\"width: 7.551em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 6.479em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1006.15em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-871\"><span class=\"mrow\" id=\"MathJax-Span-872\"><span class=\"mo\" id=\"MathJax-Span-873\" 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-874\"><span style=\"display: inline-block; position: relative; width: 4.802em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.93em, 5.92em, -999.998em); top: -4.285em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.93em, 4.289em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.464em;\"><span class=\"mtd\" id=\"MathJax-Span-875\"><span class=\"mrow\" id=\"MathJax-Span-876\"><span class=\"msubsup\" id=\"MathJax-Span-877\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-878\"><span class=\"mrow\" id=\"MathJax-Span-879\"><span class=\"mi\" id=\"MathJax-Span-880\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.865em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-881\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-888\"><span class=\"mrow\" id=\"MathJax-Span-889\"><span class=\"mn\" id=\"MathJax-Span-890\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-901\"><span class=\"mrow\" id=\"MathJax-Span-902\"><span class=\"mn\" id=\"MathJax-Span-903\" 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, 1000.93em, 6.107em, -999.998em); top: -4.471em; left: 1.913em;\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-882\"><span class=\"mrow\" id=\"MathJax-Span-883\"><span class=\"mn\" id=\"MathJax-Span-884\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.93em, 4.289em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.464em;\"><span class=\"mtd\" id=\"MathJax-Span-891\"><span class=\"mrow\" id=\"MathJax-Span-892\"><span class=\"msubsup\" id=\"MathJax-Span-893\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-894\"><span class=\"mrow\" id=\"MathJax-Span-895\"><span class=\"mi\" id=\"MathJax-Span-896\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.865em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-897\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-904\"><span class=\"mrow\" id=\"MathJax-Span-905\"><span class=\"mn\" id=\"MathJax-Span-906\" style=\"font-family: MathJax_Main;\">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(2.472em, 1000.93em, 6.386em, -999.998em); top: -4.611em; left: 3.87em;\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-885\"><span class=\"mrow\" id=\"MathJax-Span-886\"><span class=\"mn\" id=\"MathJax-Span-887\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-898\"><span class=\"mrow\" id=\"MathJax-Span-899\"><span class=\"mn\" id=\"MathJax-Span-900\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.93em, 4.336em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.464em;\"><span class=\"mtd\" id=\"MathJax-Span-907\"><span class=\"mrow\" id=\"MathJax-Span-908\"><span class=\"msubsup\" id=\"MathJax-Span-909\"><span style=\"display: inline-block; position: relative; width: 0.934em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.51em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"texatom\" id=\"MathJax-Span-910\"><span class=\"mrow\" id=\"MathJax-Span-911\"><span class=\"mi\" id=\"MathJax-Span-912\" style=\"font-family: MathJax_Main;\">a<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.865em; left: 0.515em;\"><span class=\"mn\" id=\"MathJax-Span-913\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.616em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-914\" 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-30\">\\left[\\begin{array}{ccc}  \\mathrm{a}_1 & 0 & 0 \\\\  0 & \\mathrm{a}_2 & 0 \\\\  0 & 0 & \\mathrm{a}_3  \\end{array}\\right]<\/script> = A<br> The result is true for n = 1.<br> Suppose the result for n = k then<br>\n\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-5724\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3b-18.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b) 18\" width=\"324\" height=\"282\" sizes=\"auto, (max-width: 324px) 100vw, 324px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b) 18\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b)\"><br> The result is true for n = k + 1.<br> So by the principle of mathematical induc-tion the statement is true \u2200 n \u2208 N.<\/p>\n\n<h3>Question 2.<br> If \u03b8 \u2013 \u03a6 = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-31-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-915\" style=\"width: 0.934em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 0.795em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.447em, 1000.79em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-916\"><span class=\"mfrac\" id=\"MathJax-Span-917\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.544em, 1000.42em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mi\" id=\"MathJax-Span-918\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03c0<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/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-919\" 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, 1000.51em, 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: 0.515em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.354em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-31\">\\frac{\\pi}{2}<\/script>, then show that <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-32-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-920\" style=\"width: 25.352em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 21.857em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.726em, 1021.62em, 4.522em, -999.998em); top: -3.353em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-921\"><span class=\"mrow\" id=\"MathJax-Span-922\"><span class=\"mo\" id=\"MathJax-Span-923\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-924\"><span style=\"display: inline-block; position: relative; width: 9.136em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.379em, 1004.06em, 4.942em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 4.056em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.031em, 1002.42em, 4.15em, -999.998em); top: -4.658em; left: 50%; margin-left: -1.209em;\"><span class=\"mtd\" id=\"MathJax-Span-925\"><span class=\"mrow\" id=\"MathJax-Span-926\"><span class=\"msubsup\" id=\"MathJax-Span-927\"><span style=\"display: inline-block; position: relative; width: 1.773em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1001.31em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-928\" style=\"font-family: MathJax_Main;\">cos<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 1.354em;\"><span class=\"mn\" id=\"MathJax-Span-929\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-930\"><\/span><span class=\"mi\" id=\"MathJax-Span-931\" style=\"font-family: MathJax_Math-italic; padding-left: 0.189em;\">\u03b8<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1004.06em, 4.15em, -999.998em); top: -3.213em; left: 50%; margin-left: -2.048em;\"><span class=\"mtd\" id=\"MathJax-Span-940\"><span class=\"mrow\" id=\"MathJax-Span-941\"><span class=\"mi\" id=\"MathJax-Span-942\" style=\"font-family: MathJax_Main;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-943\"><\/span><span class=\"mi\" id=\"MathJax-Span-944\" style=\"font-family: MathJax_Math-italic; padding-left: 0.189em;\">\u03b8<\/span><span class=\"mi\" id=\"MathJax-Span-945\" style=\"font-family: MathJax_Main; padding-left: 0.189em;\">sin<\/span><span class=\"mo\" id=\"MathJax-Span-946\"><\/span><span class=\"mi\" id=\"MathJax-Span-947\" style=\"font-family: MathJax_Math-italic; padding-left: 0.189em;\">\u03b8<\/span><\/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, 1004.06em, 4.942em, -999.998em); top: -4.005em; left: 5.082em;\"><span style=\"display: inline-block; position: relative; width: 4.056em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1004.06em, 4.15em, -999.998em); top: -4.658em; left: 50%; margin-left: -2.048em;\"><span class=\"mtd\" id=\"MathJax-Span-932\"><span class=\"mrow\" id=\"MathJax-Span-933\"><span class=\"mi\" id=\"MathJax-Span-934\" style=\"font-family: MathJax_Main;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-935\"><\/span><span class=\"mi\" id=\"MathJax-Span-936\" style=\"font-family: MathJax_Math-italic; padding-left: 0.189em;\">\u03b8<\/span><span class=\"mi\" id=\"MathJax-Span-937\" style=\"font-family: MathJax_Main; padding-left: 0.189em;\">sin<\/span><span class=\"mo\" id=\"MathJax-Span-938\"><\/span><span class=\"mi\" id=\"MathJax-Span-939\" style=\"font-family: MathJax_Math-italic; padding-left: 0.189em;\">\u03b8<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.985em, 1002.33em, 4.15em, -999.998em); top: -3.213em; left: 50%; margin-left: -1.163em;\"><span class=\"mtd\" id=\"MathJax-Span-948\"><span class=\"mrow\" id=\"MathJax-Span-949\"><span class=\"msubsup\" id=\"MathJax-Span-950\"><span style=\"display: inline-block; position: relative; width: 1.68em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-951\" style=\"font-family: MathJax_Main;\">sin<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 1.214em;\"><span class=\"mn\" id=\"MathJax-Span-952\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-953\"><\/span><span class=\"mi\" id=\"MathJax-Span-954\" style=\"font-family: MathJax_Math-italic; padding-left: 0.189em;\">\u03b8<\/span><\/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-955\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">]<\/span><\/span><\/span><span class=\"mrow\" id=\"MathJax-Span-956\" style=\"padding-left: 0.189em;\"><span class=\"mo\" id=\"MathJax-Span-957\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-958\"><span style=\"display: inline-block; position: relative; width: 9.648em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.379em, 1004.29em, 5.175em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 4.336em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.031em, 1002.52em, 4.336em, -999.998em); top: -4.658em; left: 50%; margin-left: -1.256em;\"><span class=\"mtd\" id=\"MathJax-Span-959\"><span class=\"mrow\" id=\"MathJax-Span-960\"><span class=\"msubsup\" id=\"MathJax-Span-961\"><span style=\"display: inline-block; position: relative; width: 1.773em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.404em, 1001.31em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-962\" style=\"font-family: MathJax_Main;\">cos<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 1.354em;\"><span class=\"mn\" id=\"MathJax-Span-963\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-964\"><\/span><span class=\"mi\" id=\"MathJax-Span-965\" style=\"font-family: MathJax_Math-italic; padding-left: 0.189em;\">\u03d5<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1004.29em, 4.336em, -999.998em); top: -3.213em; left: 50%; margin-left: -2.141em;\"><span class=\"mtd\" id=\"MathJax-Span-974\"><span class=\"mrow\" id=\"MathJax-Span-975\"><span class=\"mi\" id=\"MathJax-Span-976\" style=\"font-family: MathJax_Main;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-977\"><\/span><span class=\"mi\" id=\"MathJax-Span-978\" style=\"font-family: MathJax_Math-italic; padding-left: 0.189em;\">\u03d5<\/span><span class=\"mi\" id=\"MathJax-Span-979\" style=\"font-family: MathJax_Main; padding-left: 0.189em;\">sin<\/span><span class=\"mo\" id=\"MathJax-Span-980\"><\/span><span class=\"mi\" id=\"MathJax-Span-981\" style=\"font-family: MathJax_Math-italic; padding-left: 0.189em;\">\u03d5<\/span><\/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, 1004.29em, 5.175em, -999.998em); top: -4.005em; left: 5.315em;\"><span style=\"display: inline-block; position: relative; width: 4.336em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1004.29em, 4.336em, -999.998em); top: -4.658em; left: 50%; margin-left: -2.141em;\"><span class=\"mtd\" id=\"MathJax-Span-966\"><span class=\"mrow\" id=\"MathJax-Span-967\"><span class=\"mi\" id=\"MathJax-Span-968\" style=\"font-family: MathJax_Main;\">cos<\/span><span class=\"mo\" id=\"MathJax-Span-969\"><\/span><span class=\"mi\" id=\"MathJax-Span-970\" style=\"font-family: MathJax_Math-italic; padding-left: 0.189em;\">\u03d5<\/span><span class=\"mi\" id=\"MathJax-Span-971\" style=\"font-family: MathJax_Main; padding-left: 0.189em;\">sin<\/span><span class=\"mo\" id=\"MathJax-Span-972\"><\/span><span class=\"mi\" id=\"MathJax-Span-973\" style=\"font-family: MathJax_Math-italic; padding-left: 0.189em;\">\u03d5<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.985em, 1002.42em, 4.336em, -999.998em); top: -3.213em; left: 50%; margin-left: -1.209em;\"><span class=\"mtd\" id=\"MathJax-Span-982\"><span class=\"mrow\" id=\"MathJax-Span-983\"><span class=\"msubsup\" id=\"MathJax-Span-984\"><span style=\"display: inline-block; position: relative; width: 1.68em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-985\" style=\"font-family: MathJax_Main;\">sin<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -4.378em; left: 1.214em;\"><span class=\"mn\" id=\"MathJax-Span-986\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-987\"><\/span><span class=\"mi\" id=\"MathJax-Span-988\" style=\"font-family: MathJax_Math-italic; padding-left: 0.189em;\">\u03d5<\/span><\/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-989\" 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: -1.241em; border-left: 0px solid; width: 0px; height: 3.03em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-32\">\\left[\\begin{array}{cc}  \\cos ^2 \\theta & \\cos \\theta \\sin \\theta \\\\  \\cos \\theta \\sin \\theta & \\sin ^2 \\theta  \\end{array}\\right]\\left[\\begin{array}{cc}  \\cos ^2 \\phi & \\cos \\phi \\sin \\phi \\\\  \\cos \\phi \\sin \\phi & \\sin ^2 \\phi  \\end{array}\\right]<\/script> = 0<\/h3>\n\n<p>Answer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-5723\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3b-19.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b) 19\" width=\"359\" height=\"643\" sizes=\"auto, (max-width: 359px) 100vw, 359px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b) 19\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b)\"><\/p>\n\n<h3>Question 3.<br> If A = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-33-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-990\" 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-991\"><span class=\"mrow\" id=\"MathJax-Span-992\"><span class=\"mo\" id=\"MathJax-Span-993\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-994\"><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-995\"><span class=\"mrow\" id=\"MathJax-Span-996\"><span class=\"mn\" id=\"MathJax-Span-997\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.26em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1002\"><span class=\"mrow\" id=\"MathJax-Span-1003\"><span class=\"mn\" id=\"MathJax-Span-1004\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1001.26em, 4.988em, -999.998em); top: -4.005em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.26em, 4.243em, -999.998em); top: -4.658em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-998\"><span class=\"mrow\" id=\"MathJax-Span-999\"><span class=\"mo\" id=\"MathJax-Span-1000\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-1001\" style=\"font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -3.26em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1005\"><span class=\"mrow\" id=\"MathJax-Span-1006\"><span class=\"mo\" id=\"MathJax-Span-1007\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-1008\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-1009\" 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-33\">\\left[\\begin{array}{rr}  3 & -4 \\\\  1 & -1  \\end{array}\\right]<\/script>, then show that A<sup>n<\/sup> = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-34-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-1010\" style=\"width: 9.462em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 8.157em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.773em, 1007.92em, 4.429em, -999.998em); top: -3.353em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-1011\"><span class=\"mrow\" id=\"MathJax-Span-1012\"><span class=\"mo\" id=\"MathJax-Span-1013\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-1014\"><span style=\"display: inline-block; position: relative; width: 6.712em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.565em, 1002.85em, 4.895em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 2.845em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1002.85em, 4.243em, -999.998em); top: -4.658em; left: 50%; margin-left: -1.442em;\"><span class=\"mtd\" id=\"MathJax-Span-1015\"><span class=\"mrow\" id=\"MathJax-Span-1016\"><span class=\"mn\" id=\"MathJax-Span-1017\" style=\"font-family: MathJax_Main;\">1<\/span><span class=\"mo\" id=\"MathJax-Span-1018\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">+<\/span><span class=\"mn\" id=\"MathJax-Span-1019\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">2<\/span><span class=\"mi\" id=\"MathJax-Span-1020\" style=\"font-family: MathJax_Math-italic;\">n<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.451em, 1000.56em, 4.15em, -999.998em); top: -3.26em; left: 50%; margin-left: -0.277em;\"><span class=\"mtd\" id=\"MathJax-Span-1026\"><span class=\"mrow\" id=\"MathJax-Span-1027\"><span class=\"mi\" id=\"MathJax-Span-1028\" style=\"font-family: MathJax_Math-italic;\">n<\/span><\/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, 1002.85em, 4.988em, -999.998em); top: -4.005em; left: 3.87em;\"><span style=\"display: inline-block; position: relative; width: 2.845em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1001.87em, 4.243em, -999.998em); top: -4.658em; left: 50%; margin-left: -0.93em;\"><span class=\"mtd\" id=\"MathJax-Span-1021\"><span class=\"mrow\" id=\"MathJax-Span-1022\"><span class=\"mo\" id=\"MathJax-Span-1023\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-1024\" style=\"font-family: MathJax_Main;\">4<\/span><span class=\"mi\" id=\"MathJax-Span-1025\" style=\"font-family: MathJax_Math-italic;\">n<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1002.85em, 4.243em, -999.998em); top: -3.26em; left: 50%; margin-left: -1.442em;\"><span class=\"mtd\" id=\"MathJax-Span-1029\"><span class=\"mrow\" id=\"MathJax-Span-1030\"><span class=\"mn\" id=\"MathJax-Span-1031\" style=\"font-family: MathJax_Main;\">1<\/span><span class=\"mo\" id=\"MathJax-Span-1032\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-1033\" style=\"font-family: MathJax_Main; padding-left: 0.235em;\">2<\/span><span class=\"mi\" id=\"MathJax-Span-1034\" style=\"font-family: MathJax_Math-italic;\">n<\/span><\/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-1035\" 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-34\">\\left[\\begin{array}{cc}  1+2 n & -4 n \\\\  n & 1-2 n  \\end{array}\\right]<\/script> for any integer n \u2265 1, by using mathematical induction.<\/h3>\n\n<p>Answer:<br> We shall prove the result by mathematical induction.<br>\n\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-5721\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3b-20.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b) 20\" width=\"288\" height=\"187\" data-pin-nopin=\"true\"><\/p>\n\n<br> \u2234 The given result is true for n = k + 1<br> \u2234 By Mathematical induction the given result is true for all positive integral values of n.<\/p>\n\n<h3>Question 4.<br> Given examples of two square matrices A and B of the same order for which AB = 0 but BA \u2260 0.<\/h3>\n\n<p>Answer:<br>\n\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-5721\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3b-21.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b) 21\" width=\"288\" height=\"187\" data-pin-nopin=\"true\"><\/p>\n\n<h3>Question 5.<br> A trust fund has to invest Rs. 30,000 in two different types of bonds. The first bond pays 5% interest per year, and the second bond pays 7% interest per year. Using matrix multiplication determine how to divide Rs. 30,000 among the two types of bonds, if the trust fund must obtain an annual total interest of (a) Rs. 1800 and (b) Rs. 2,000.<\/h3>\n\n<p>Answer:<br> Let the first bond be \u2018x\u2019, then the second bond will be 30,000 \u2013 x.<br> Rate of interest are 5% and 7% means 0.05 and 0.07.<br> a) [x 30,000 \u2013 x]<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-35-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-1036\" style=\"width: 3.73em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.218em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.773em, 1002.99em, 4.429em, -999.998em); top: -3.353em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-1037\"><span class=\"mrow\" id=\"MathJax-Span-1038\"><span class=\"mo\" id=\"MathJax-Span-1039\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-1040\"><span style=\"display: inline-block; position: relative; width: 1.773em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.565em, 1001.77em, 4.942em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 1.773em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.73em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1041\"><span class=\"mrow\" id=\"MathJax-Span-1042\"><span class=\"mn\" id=\"MathJax-Span-1043\" style=\"font-family: MathJax_Main;\">0.05<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1001.77em, 4.15em, -999.998em); top: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1044\"><span class=\"mrow\" id=\"MathJax-Span-1045\"><span class=\"mn\" id=\"MathJax-Span-1046\" style=\"font-family: MathJax_Main;\">0.07<\/span><\/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-1047\" 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-35\">\\left[\\begin{array}{l}  0.05 \\\\  0.07  \\end{array}\\right]<\/script> = [1800]<br> \u21d2 0.05x + 0.07 (30,000 \u2013 x) = 1800<br> \u21d2 \u2013 0.02x + 0.07 (30,000) = 1800<br> \u21d2 \u2013 0.02x + 2100 = 1800<br> \u21d2 \u2013 0.02x = \u2013 300<br> \u21d2 x = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-36-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-1048\" style=\"width: 1.959em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.68em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1001.68em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-1049\"><span class=\"mfrac\" id=\"MathJax-Span-1050\"><span style=\"display: inline-block; position: relative; width: 1.4em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1001.03em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.51em;\"><span class=\"mn\" id=\"MathJax-Span-1051\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">300<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1001.21em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.603em;\"><span class=\"mn\" id=\"MathJax-Span-1052\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">0.02<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.4em, 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: 1.4em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-36\">\\frac{300}{0.02}<\/script> = 300 \u00d7 <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-37-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-1053\" style=\"width: 1.68em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.447em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1001.45em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-1054\"><span class=\"mfrac\" id=\"MathJax-Span-1055\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1001.03em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.51em;\"><span class=\"mn\" id=\"MathJax-Span-1056\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">100<\/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-1057\" 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, 1001.17em, 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: 1.167em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-37\">\\frac{100}{2}<\/script> = 15, 000<br> First bond = 15, 000<br> Second bond = 30,000 \u2013 x<br> = 30,000 \u2013 15,000 = 15,000<\/p><p>(b) [x 30,000 \u2013 x]<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-38-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-1058\" style=\"width: 3.73em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.218em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.773em, 1002.99em, 4.429em, -999.998em); top: -3.353em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-1059\"><span class=\"mrow\" id=\"MathJax-Span-1060\"><span class=\"mo\" id=\"MathJax-Span-1061\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-1062\"><span style=\"display: inline-block; position: relative; width: 1.773em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.565em, 1001.77em, 4.942em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 1.773em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.73em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1063\"><span class=\"mrow\" id=\"MathJax-Span-1064\"><span class=\"mn\" id=\"MathJax-Span-1065\" style=\"font-family: MathJax_Main;\">0.05<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1001.77em, 4.15em, -999.998em); top: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-1066\"><span class=\"mrow\" id=\"MathJax-Span-1067\"><span class=\"mn\" id=\"MathJax-Span-1068\" style=\"font-family: MathJax_Main;\">0.07<\/span><\/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-1069\" 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-38\">\\left[\\begin{array}{l}  0.05 \\\\  0.07  \\end{array}\\right]<\/script> = [2000]<br> \u21d2 0.05x + 0.07 (30,000 \u2013 x) = 2000<br> \u21d2 \u2013 0.02x + 0.07 (30,000) = 2000<br> \u21d2 \u2013 0.02x + 2100 = 2000<br> \u21d2 x = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-39-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-1070\" style=\"width: 1.959em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.68em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1001.68em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-1071\"><span class=\"mfrac\" id=\"MathJax-Span-1072\"><span style=\"display: inline-block; position: relative; width: 1.4em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1001.03em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.51em;\"><span class=\"mn\" id=\"MathJax-Span-1073\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">100<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1001.21em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.603em;\"><span class=\"mn\" id=\"MathJax-Span-1074\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">0.02<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.4em, 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: 1.4em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-39\">\\frac{100}{0.02}<\/script> = 5, 000<br> \u2234 Second bond = 30,000 \u2013 x<br> = 30,000 \u2013 5,000<br> = 25,000<\/p><\/div>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>TS Inter 1st Year Maths 1A Matrices Solutions Exercise 3(b) I. Question 1. Find the following products wherever possible. i) [-1 4 2]\u23a1\u23a3\u23a2513\u23a4\u23a6\u23a5 Answer: It is a product of 1 \u00d7 3 and 3 \u00d7 1 matrices and the resulting is an 1 \u00d7 1 matrix. [-1 4 2]\u23a1\u23a3\u23a2513\u23a4\u23a6\u23a5 = [-1 \u00d7 5 + 4 [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":2665,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[1544,15,38],"tags":[1574,1570,1551,1572,1573,1563,1553,1571,1559,1548,1561,1565,1568,1562,1569,1566,1567,1564],"class_list":{"0":"post-2656","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-study-material-tg-inter","8":"category-telangana","9":"category-tg-inter","10":"tag-inter-1st-year-maths-1a-matrices-important-questions","11":"tag-inter-1st-year-maths-1a-matrices-questions-answers","12":"tag-inter-1st-year-maths-1a-matrices-solutions","13":"tag-maths-1a-chapter-3-matrices-short-questions-answers","14":"tag-maths-1a-chapter-3-matrices-solved-problems","15":"tag-maths-1a-matrices-solutions-ts-inter","16":"tag-maths-1a-matrices-step-by-step-solutions-ts-inter","17":"tag-matrices-class-11-ts-intermediate-maths","18":"tag-ts-inter-1st-year-maths-1a-chapter-3-matrices-solutions-pdf-download","19":"tag-ts-inter-1st-year-maths-1a-matrices-pdf","20":"tag-ts-inter-1st-year-maths-1a-solutions-chapter-3","21":"tag-ts-inter-maths-1a-chapter-3-exercise-solutions","22":"tag-ts-inter-maths-1a-chapter-3-matrices-exercise-3-1-solutions","23":"tag-ts-inter-maths-1a-chapter-3-matrices-solutions","24":"tag-ts-inter-maths-1a-matrices-solved-examples","25":"tag-ts-inter-maths-1a-textbook-solutions-chapter-3","26":"tag-ts-intermediate-1st-year-maths-matrices-solutions","27":"tag-ts-intermediate-maths-1a-chapter-3-solutions"},"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.3 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b)<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3b\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(b)\" \/>\n<meta property=\"og:description\" content=\"TS Inter 1st Year Maths 1A Matrices Solutions Exercise 3(b) I. Question 1. Find the following products wherever possible. i) [-1 4 2]\u23a1\u23a3\u23a2513\u23a4\u23a6\u23a5 Answer: It is a product of 1 \u00d7 3 and 3 \u00d7 1 matrices and the resulting is an 1 \u00d7 1 matrix. 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