{"id":4483,"date":"2016-05-16T11:49:00","date_gmt":"2016-05-16T06:19:00","guid":{"rendered":"http:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-3-3\/"},"modified":"2018-05-08T10:04:32","modified_gmt":"2018-05-08T04:34:32","slug":"ncert-solutions-class-12-maths-exercise-3-3","status":"publish","type":"post","link":"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-3-3\/","title":{"rendered":"NCERT Solutions class 12 Maths Exercise 3.3"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_76 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 eztoc-toggle-hide-by-default' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-3-3\/#NCERT_Solutions_class_12_Maths_Matrices\" >NCERT Solutions class 12 Maths Matrices\u00a0\u00a0<\/a><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-3-3\/#NCERT_Solutions_class_12_Maths_Exercise_33\" >NCERT Solutions class 12 Maths Exercise 3.3<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-3-3\/#3_If_A_and_B_then_verify_that\" >3. If A\u2019 = \u00a0and B = \u00a0then verify that:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-3-3\/#NCERT_Solutions_class_12_Maths_Exercise_33-2\" >NCERT Solutions class 12 Maths Exercise 3.3<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-3-3\/#4_If_A_and_B_then_find_A_2B\" >4. If A\u2019 = \u00a0and B = \u00a0then find (A + 2B)\u2019.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-3-3\/#NCERT_Solutions_class_12_Maths_Exercise_33-3\" >NCERT Solutions class 12 Maths Exercise 3.3<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-3-3\/#5_For_the_matrices_A_and_B_verify_that_AB_BA_where\" >5. For the matrices A and B, verify that (AB)\u2019 = B\u2019A\u2019, where:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-3-3\/#NCERT_Solutions_class_12_Maths_Exercise_33-4\" >NCERT Solutions class 12 Maths Exercise 3.3<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-3-3\/#6_i_If_A_then_verify_that_AA_I\" >6. (i) If A = \u00a0then verify that A\u2019A = I.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-3-3\/#NCERT_Solutions_class_12_Maths_Exercise_33-5\" >NCERT Solutions class 12 Maths Exercise 3.3<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-3-3\/#7_i_Show_that_the_matrix_A_is_a_symmetric_matrix\" >7. (i) Show that the matrix A = \u00a0is a symmetric matrix.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-3-3\/#8_For_a_matrix_A_verify_that\" >8. For a matrix A = \u00a0verify that:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-3-3\/#9_Find_A_A_and_A_%E2%80%93_A_when_A\" >9. Find \u00a0(A + A\u2019) and \u00a0(A \u2013 A\u2019) when A = \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-3-3\/#NCERT_Solutions_class_12_Maths_Exercise_33-6\" >NCERT Solutions class 12 Maths Exercise 3.3<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-3-3\/#10_Express_the_following_matrices_as_the_sum_of_a_symmetric_and_skew_symmetric_matrix\" >10. Express the following matrices as the sum of a symmetric and skew symmetric matrix:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-16\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-3-3\/#NCERT_Solutions_class_12_Maths_Exercise_33-7\" >NCERT Solutions class 12 Maths Exercise 3.3<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-17\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-3-3\/#Choose_the_correct_answer_in_Exercises_11_and_12\" >Choose the correct answer in Exercises 11 and 12.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-18\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-3-3\/#NCERT_Solutions_class_12_Maths_Exercise_33-8\" >NCERT Solutions class 12 Maths Exercise 3.3<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-19\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-3-3\/#12_If_A_then_A_A_I_if_the_value_of_is\" >12. If A = , then A + A\u2019 = I, if the value of \u00a0is:<\/a><\/li><\/ul><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-20\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-3-3\/#NCERT_Solutions_class_12_Maths_Exercise_33-9\" >NCERT Solutions class 12 Maths Exercise 3.3<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-21\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-3-3\/#CBSE_App_for_Students\" >CBSE App for Students<\/a><\/li><\/ul><\/nav><\/div>\n<p>NCERT Solutions class 12 Maths Exercise 3.3 Class 12 Maths book solutions are available in PDF format for free download. These ncert book chapter wise questions and answers are very helpful for CBSE board exam. CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. Class 12 Maths chapter wise NCERT solution for Maths part 1 and Maths part 2 for all the chapters can be downloaded from our website and myCBSEguide mobile app for free.<\/p>\n<p><strong>Download\u00a0<a href=\"https:\/\/mycbseguide.com\/downloads\/cbse-class-12-mathematics-matrices\/1287\/ncert-solutions\/5\/\">NCERT solutions for\u00a0Matrices\u00a0<\/a>as PDF.<\/strong><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/media-mycbseguide.s3.ap-south-1.amazonaws.com\/images\/blog\/12%20Maths%20Book%20pic.jpg\" alt=\"NCERT Solutions class 12 Maths Exercise 3.3\" width=\"185\" height=\"241\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Matrices\"><\/span>NCERT Solutions class 12 Maths Matrices<strong>\u00a0<\/strong><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p style=\"text-align: justify;\"><strong>1. Find the transpose of each of the following matrices:<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(i) <img decoding=\"async\" style=\"height: 83px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image001.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(ii) <img decoding=\"async\" style=\"height: 49px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image002.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(iii) <img decoding=\"async\" style=\"height: 81px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image003.png\" \/>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. (i)<\/strong> Let A = <img decoding=\"async\" style=\"height: 83px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image001.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image004.png\" \/>\u00a0 Transpose of A = A\u2019 or A<sup>T<\/sup> = <img decoding=\"async\" style=\"height: 45px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image005.png\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>(ii)<\/strong> <img decoding=\"async\" style=\"height: 49px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image002.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image004.png\" \/>\u00a0 Transpose of A = A\u2019 or A<sup>T<\/sup> = <img decoding=\"async\" style=\"height: 49px; width: 63px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image006.png\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>(iii)<\/strong> <img decoding=\"async\" style=\"height: 81px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image003.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image004.png\" \/>\u00a0 Transpose of A = A\u2019 or A<sup>T<\/sup> = <img decoding=\"async\" style=\"height: 81px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image007.png\" \/><\/p>\n<hr \/>\n<p style=\"text-align: justify;\"><strong>2. If A = <img decoding=\"async\" style=\"height: 75px; width: 88px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image008.png\" \/>\u00a0and B = <img decoding=\"async\" style=\"height: 75px; width: 105px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image009.png\" \/>\u00a0then verify that:<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(i) <img decoding=\"async\" style=\"height: 28px; width: 125px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image010.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(ii) <img decoding=\"async\" style=\"height: 28px; width: 136px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image011.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. (i)<\/strong> A + B = <img decoding=\"async\" style=\"height: 75px; width: 196px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image012.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 75px; width: 151px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image013.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 75px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image014.png\" \/><\/p>\n<p style=\"text-align: justify;\">L.H.S. = (A + B)\u2019 = <img decoding=\"async\" style=\"height: 77px; width: 106px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image015.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 75px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image016.png\" \/><\/p>\n<p style=\"text-align: justify;\">R.H.S. = A\u2019 + B\u2019 = <img decoding=\"async\" style=\"height: 77px; width: 202px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image017.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 75px; width: 199px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image018.png\" \/><\/p>\n<p style=\"text-align: justify;\">\u00a0= <img decoding=\"async\" style=\"height: 75px; width: 159px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image019.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 75px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image016.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image004.png\" \/>\u00a0 L.H.S. = R.H.S.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Proved.<\/p>\n<p style=\"text-align: justify;\"><strong>(ii)<\/strong> A \u2013 B = <img decoding=\"async\" style=\"height: 75px; width: 195px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image020.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 75px; width: 151px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image021.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 75px; width: 105px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image022.png\" \/><\/p>\n<p style=\"text-align: justify;\">L.H.S. = (A \u2013 B)\u2019 = <img decoding=\"async\" style=\"height: 77px; width: 106px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image023.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 75px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image024.png\" \/><\/p>\n<p style=\"text-align: justify;\">R.H.S. = A\u2019 \u2013 B\u2019 = <img decoding=\"async\" style=\"height: 77px; width: 202px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image025.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 75px; width: 199px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image026.png\" \/><\/p>\n<p style=\"text-align: justify;\">\u00a0 = <img decoding=\"async\" style=\"height: 75px; width: 159px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image027.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 75px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image024.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image004.png\" \/>\u00a0 L.H.S. = R.H.S.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Proved.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_33\"><\/span>NCERT Solutions class 12 Maths Exercise 3.3<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"3_If_A_and_B_then_verify_that\"><\/span><strong>3. If A\u2019 = <img decoding=\"async\" style=\"height: 75px; width: 55px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image028.png\" \/>\u00a0and B = <img decoding=\"async\" style=\"height: 49px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image029.png\" \/>\u00a0then verify that:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>(i) <img decoding=\"async\" style=\"height: 28px; width: 125px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image010.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(ii) <img decoding=\"async\" style=\"height: 28px; width: 136px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image011.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: A\u2019 = <img decoding=\"async\" style=\"height: 75px; width: 55px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image028.png\" \/>\u00a0and B = <img decoding=\"async\" style=\"height: 49px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image029.png\" \/>\u00a0then (A\u2019)\u2019 = A = <img decoding=\"async\" style=\"height: 49px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image030.png\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>(i)<\/strong> A + B = <img decoding=\"async\" style=\"height: 49px; width: 169px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image031.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 49px; width: 224px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image032.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image004.png\" \/>\u00a0 L.H.S. = (A + B)\u2019 = <img decoding=\"async\" style=\"height: 75px; width: 128px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image033.png\" \/><\/p>\n<p style=\"text-align: justify;\">R.H.S. = A\u2019 + B\u2019 = <img decoding=\"async\" style=\"height: 75px; width: 142px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image034.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 75px; width: 121px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image035.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" style=\"height: 75px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image036.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 75px; width: 46px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image037.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image004.png\" \/>\u00a0 L.H.S. = R.H.S.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Proved.<\/p>\n<p style=\"text-align: justify;\"><strong>(ii)<\/strong> A \u2013 B = <img decoding=\"async\" style=\"height: 49px; width: 169px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image038.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 49px; width: 252px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image039.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image004.png\" \/>\u00a0 L.H.S. = (A \u2013 B)\u2019 = <img decoding=\"async\" style=\"height: 75px; width: 179px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image040.png\" \/><\/p>\n<p style=\"text-align: justify;\">R.H.S. = A\u2019 \u2013 B\u2019 = <img decoding=\"async\" style=\"height: 75px; width: 142px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image041.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 75px; width: 121px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image042.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" style=\"height: 75px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image043.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 75px; width: 71px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image044.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image004.png\" \/>\u00a0 L.H.S. = R.H.S.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Proved.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_33-2\"><\/span>NCERT Solutions class 12 Maths Exercise 3.3<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"4_If_A_and_B_then_find_A_2B\"><\/span><strong>4. If A\u2019 = <img decoding=\"async\" style=\"height: 49px; width: 55px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image045.png\" \/>\u00a0and B = <img decoding=\"async\" style=\"height: 49px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image046.png\" \/>\u00a0then find (A + 2B)\u2019.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: A\u2019 = <img decoding=\"async\" style=\"height: 49px; width: 55px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image045.png\" \/>\u00a0and B = <img decoding=\"async\" style=\"height: 49px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image046.png\" \/>\u00a0then (A\u2019)\u2019 = A = <img decoding=\"async\" style=\"height: 49px; width: 55px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image047.png\" \/><\/p>\n<p style=\"text-align: justify;\">A +2B = <img decoding=\"async\" style=\"height: 49px; width: 131px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image048.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 49px; width: 121px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image049.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 49px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image050.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 49px; width: 55px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image051.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image004.png\" \/>\u00a0 (A + 2B)\u2019 = <img decoding=\"async\" style=\"height: 52px; width: 127px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image052.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_33-3\"><\/span>NCERT Solutions class 12 Maths Exercise 3.3<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"5_For_the_matrices_A_and_B_verify_that_AB_BA_where\"><\/span><strong>5. For the matrices A and B, verify that (AB)\u2019 = B\u2019A\u2019, where:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>(i) A = <img decoding=\"async\" style=\"height: 75px; width: 42px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image053.png\" \/>\u00a0B = <img decoding=\"async\" style=\"height: 28px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image054.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(ii) A = <img decoding=\"async\" style=\"height: 75px; width: 33px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image055.png\" \/>\u00a0B = <img decoding=\"async\" style=\"height: 28px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image056.png\" \/>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. (i)<\/strong> AB = <img decoding=\"async\" style=\"height: 75px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image057.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 75px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image058.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image004.png\" \/>\u00a0 L.H.S. = (AB)\u2019 = <img decoding=\"async\" style=\"height: 77px; width: 108px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image059.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 75px; width: 106px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image060.png\" \/><\/p>\n<p style=\"text-align: justify;\">R.H.S. = B\u2019A\u2019 = <img decoding=\"async\" style=\"height: 77px; width: 108px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image061.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 75px; width: 108px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image062.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 75px; width: 106px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image060.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image004.png\" \/>\u00a0 L.H.S. = R.H.S.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Proved.<\/p>\n<p style=\"text-align: justify;\"><strong>(ii)<\/strong> AB = <img decoding=\"async\" style=\"height: 75px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image063.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 75px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image064.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image004.png\" \/>\u00a0 L.H.S. = (AB)\u2019 = <img decoding=\"async\" style=\"height: 77px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image065.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 75px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image066.png\" \/><\/p>\n<p style=\"text-align: justify;\">R.H.S. = B\u2019A\u2019 = <img decoding=\"async\" style=\"height: 77px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image067.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 75px; width: 88px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image068.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 75px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image066.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image004.png\" \/>\u00a0 L.H.S. = R.H.S.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Proved.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_33-4\"><\/span>NCERT Solutions class 12 Maths Exercise 3.3<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"6_i_If_A_then_verify_that_AA_I\"><\/span><strong>6. (i) If A = <img decoding=\"async\" style=\"height: 49px; width: 121px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image069.png\" \/>\u00a0then verify that A\u2019A = I.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>(ii) If A = <img decoding=\"async\" style=\"height: 49px; width: 121px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image070.png\" \/>\u00a0then verify that A\u2019A = I.<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. (i)<\/strong> L.H.S. = A\u2019A = <img decoding=\"async\" style=\"height: 52px; width: 232px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image071.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" style=\"height: 49px; width: 228px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image072.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" style=\"height: 53px; width: 329px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image073.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 49px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image074.png\" \/>\u00a0= I = R.H.S.<\/p>\n<p style=\"text-align: justify;\"><strong>(ii)<\/strong> L.H.S. = A\u2019A = <img decoding=\"async\" style=\"height: 52px; width: 232px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image075.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 49px; width: 228px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image076.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" style=\"height: 53px; width: 332px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image077.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 49px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image074.png\" \/>\u00a0= I = R.H.S.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_33-5\"><\/span>NCERT Solutions class 12 Maths Exercise 3.3<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"7_i_Show_that_the_matrix_A_is_a_symmetric_matrix\"><\/span><strong>7. (i) Show that the matrix A = <img decoding=\"async\" style=\"height: 75px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image078.png\" \/>\u00a0is a symmetric matrix.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>(ii) Show that the matrix A = <img decoding=\"async\" style=\"height: 75px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image079.png\" \/>\u00a0is a skew symmetric matrix.<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. (i)<\/strong> Given: A = <img decoding=\"async\" style=\"height: 75px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image078.png\" \/>\u00a0\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\">Changing rows of matrix A as the columns of new matrix A\u2019 = <img decoding=\"async\" style=\"height: 75px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image078.png\" \/>\u00a0= A<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image004.png\" \/>\u00a0 A\u2019 = A<\/p>\n<p style=\"text-align: justify;\">Therefore, by definitions of symmetric matrix, A is a symmetric matrix.<\/p>\n<p style=\"text-align: justify;\"><strong>(ii)<\/strong> Given: A = <img decoding=\"async\" style=\"height: 75px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image079.png\" \/>\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image004.png\" \/>\u00a0 A\u2019 = <img decoding=\"async\" style=\"height: 77px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image080.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 75px; width: 96px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image081.png\" \/><\/p>\n<p style=\"text-align: justify;\">Taking <img decoding=\"async\" style=\"height: 28px; width: 33px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image082.png\" \/>\u00a0common, A\u2019 = <img decoding=\"async\" style=\"height: 75px; width: 111px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image083.png\" \/>\u00a0= \u2013 A\u00a0\u00a0 [From eq. (i)]\n<p style=\"text-align: justify;\">Therefore, by definition matrix A is a skew-symmetric matrix<\/p>\n<hr \/>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"8_For_a_matrix_A_verify_that\"><\/span><strong>8. For a matrix A = <img decoding=\"async\" style=\"height: 49px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image084.png\" \/>\u00a0verify that:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>(i) (A + A\u2019) is a symmetric matrix.<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(ii) (A \u2013 A\u2019) is a skew symmetric matrix.<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. (i)<\/strong> Given: A = <img decoding=\"async\" style=\"height: 49px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image085.png\" \/><\/p>\n<p style=\"text-align: justify;\">Let B = A + A\u2019 = <img decoding=\"async\" style=\"height: 52px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image086.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 49px; width: 105px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image087.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 49px; width: 165px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image088.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image004.png\" \/>\u00a0 B\u2019 = <img decoding=\"async\" style=\"height: 52px; width: 140px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image089.png\" \/>\u00a0= B<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image004.png\" \/>\u00a0 B = A + A\u2019 is a symmetric matrix.<\/p>\n<p style=\"text-align: justify;\"><strong>(ii)<\/strong> Given: <img decoding=\"async\" style=\"height: 49px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image085.png\" \/><\/p>\n<p style=\"text-align: justify;\">Let B = A \u2013 A\u2019 = <img decoding=\"async\" style=\"height: 52px; width: 106px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image090.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 49px; width: 105px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image091.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 49px; width: 165px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image092.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image004.png\" \/>\u00a0 B\u2019 = <img decoding=\"async\" style=\"height: 52px; width: 138px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image093.png\" \/><\/p>\n<p style=\"text-align: justify;\">Taking <img decoding=\"async\" style=\"height: 28px; width: 33px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image082.png\" \/>\u00a0common, <img decoding=\"async\" style=\"height: 49px; width: 72px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image094.png\" \/>= \u2013 B<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image004.png\" \/>\u00a0 B = A \u2013 A\u2019 is a skew-symmetric matrix.<\/p>\n<hr \/>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"9_Find_A_A_and_A_%E2%80%93_A_when_A\"><\/span><strong>9. Find <img decoding=\"async\" style=\"height: 42px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image095.png\" \/>\u00a0(A + A\u2019) and <img decoding=\"async\" style=\"height: 42px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image095.png\" \/>\u00a0(A \u2013 A\u2019) when A = <img decoding=\"async\" style=\"height: 75px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image096.png\" \/>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: A = <img decoding=\"async\" style=\"height: 75px; width: 96px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image097.png\" \/>\u00a0\u00a0\u00a0<img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image004.png\" \/>\u00a0\u00a0A\u2019 = <img decoding=\"async\" style=\"height: 75px; width: 106px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image098.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now, A + A\u2019 = <img decoding=\"async\" style=\"height: 75px; width: 211px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image099.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 75px; width: 161px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image100.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 75px; width: 66px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image101.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image004.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 42px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image095.png\" \/>\u00a0(A + A\u2019) = <img decoding=\"async\" style=\"height: 75px; width: 157px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image102.png\" \/><\/p>\n<p style=\"text-align: justify;\">Now, A \u2013 A\u2019 = <img decoding=\"async\" style=\"height: 75px; width: 211px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image103.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 75px; width: 161px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image104.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 75px; width: 124px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image105.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image004.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 42px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image095.png\" \/>\u00a0(A \u2013 A\u2019) = <img decoding=\"async\" style=\"height: 75px; width: 136px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image106.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 75px; width: 106px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image107.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_33-6\"><\/span>NCERT Solutions class 12 Maths Exercise 3.3<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"10_Express_the_following_matrices_as_the_sum_of_a_symmetric_and_skew_symmetric_matrix\"><\/span><strong>10. Express the following matrices as the sum of a symmetric and skew symmetric matrix:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>(i) <img decoding=\"async\" style=\"height: 49px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image108.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(ii) <img decoding=\"async\" style=\"height: 75px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image109.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(iii) <img decoding=\"async\" style=\"height: 75px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image110.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(iv) <img decoding=\"async\" style=\"height: 49px; width: 55px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image111.png\" \/>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. (i)<\/strong> Given: \u00a0A = <img decoding=\"async\" style=\"height: 49px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image108.png\" \/>\u00a0\u00a0\u00a0\u00a0<img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image004.png\" \/>\u00a0\u00a0A\u2019 = <img decoding=\"async\" style=\"height: 49px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image112.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image004.png\" \/>\u00a0 Symmetric matrix = <img decoding=\"async\" style=\"height: 42px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image095.png\" \/>\u00a0(A + A\u2019) = <img decoding=\"async\" style=\"height: 51px; width: 152px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image113.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" style=\"height: 49px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image114.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 49px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image115.png\" \/><\/p>\n<p style=\"text-align: justify;\">And Skew symmetric matrix = <img decoding=\"async\" style=\"height: 42px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image095.png\" \/>\u00a0(A \u2013 A\u2019) = <img decoding=\"async\" style=\"height: 51px; width: 152px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image116.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" style=\"height: 49px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image117.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 49px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image118.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image004.png\" \/>\u00a0Given matrix A is sum of Symmetric matrix <img decoding=\"async\" style=\"height: 49px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image115.png\" \/>\u00a0and Skew symmetric matrix <img decoding=\"async\" style=\"height: 49px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image118.png\" \/>.<\/p>\n<p style=\"text-align: justify;\"><strong>(ii)<\/strong> Given: A = <img decoding=\"async\" style=\"height: 75px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image109.png\" \/>\u00a0\u00a0\u00a0<img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image004.png\" \/>\u00a0\u00a0A\u2019 = <img decoding=\"async\" style=\"height: 75px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image109.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image004.png\" \/>\u00a0 Symmetric matrix = <img decoding=\"async\" style=\"height: 42px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image095.png\" \/>\u00a0(A + A\u2019) = <img decoding=\"async\" style=\"height: 77px; width: 240px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image119.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" style=\"height: 75px; width: 116px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image120.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 75px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image109.png\" \/><\/p>\n<p style=\"text-align: justify;\">And Skew symmetric matrix = <img decoding=\"async\" style=\"height: 42px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image095.png\" \/>\u00a0(A \u2013 A\u2019) = <img decoding=\"async\" style=\"height: 77px; width: 240px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image121.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" style=\"height: 75px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image122.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 75px; width: 66px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image101.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image004.png\" \/>\u00a0Given matrix A is sum of Symmetric matrix <img decoding=\"async\" style=\"height: 75px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image109.png\" \/>\u00a0and Skew symmetric\u00a0 matrix <img decoding=\"async\" style=\"height: 75px; width: 66px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image101.png\" \/>.<\/p>\n<p style=\"text-align: justify;\"><strong>(iii)<\/strong> Given: A = <img decoding=\"async\" style=\"height: 75px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image110.png\" \/>\u00a0\u00a0\u00a0<img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image004.png\" \/>\u00a0\u00a0A\u2019 = <img decoding=\"async\" style=\"height: 75px; width: 113px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image123.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image004.png\" \/>\u00a0 Symmetric matrix = <img decoding=\"async\" style=\"height: 42px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image095.png\" \/>\u00a0(A + A\u2019) = <img decoding=\"async\" style=\"height: 77px; width: 252px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image124.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" style=\"height: 75px; width: 124px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image125.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 99px; width: 142px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image126.png\" \/><\/p>\n<p style=\"text-align: justify;\">And Skew symmetric matrix = <img decoding=\"async\" style=\"height: 42px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image095.png\" \/>\u00a0(A \u2013 A\u2019) = <img decoding=\"async\" style=\"height: 77px; width: 252px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image127.png\" \/><\/p>\n<p style=\"text-align: justify;\">= <img decoding=\"async\" style=\"height: 75px; width: 109px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image128.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 99px; width: 140px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image129.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image004.png\" \/>\u00a0Given matrix A is sum of Symmetric matrix <img decoding=\"async\" style=\"height: 99px; width: 142px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image126.png\" \/>\u00a0and Skew symmetric\u00a0 matrix <img decoding=\"async\" style=\"height: 99px; width: 140px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image129.png\" \/>.<\/p>\n<p style=\"text-align: justify;\"><strong>(iv)<\/strong> Given: A = <img decoding=\"async\" style=\"height: 49px; width: 55px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image111.png\" \/>\u00a0\u00a0\u00a0\u00a0<img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image004.png\" \/>\u00a0\u00a0A\u2019 = <img decoding=\"async\" style=\"height: 49px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image130.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image004.png\" \/>\u00a0 Symmetric matrix = <img decoding=\"async\" style=\"height: 42px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image095.png\" \/>\u00a0(A + A\u2019) = <img decoding=\"async\" style=\"height: 51px; width: 151px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image131.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 49px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image132.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 49px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image133.png\" \/><\/p>\n<p style=\"text-align: justify;\">And Skew symmetric matrix = <img decoding=\"async\" style=\"height: 42px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image095.png\" \/>\u00a0(A \u2013 A\u2019) = <img decoding=\"async\" style=\"height: 51px; width: 149px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image134.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 49px; width: 55px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image135.png\" \/><\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image004.png\" \/>\u00a0Given matrix A is sum of Symmetric matrix <img decoding=\"async\" style=\"height: 49px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image133.png\" \/>\u00a0and Skew symmetric matrix <img decoding=\"async\" style=\"height: 49px; width: 55px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image135.png\" \/>.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_33-7\"><\/span>NCERT Solutions class 12 Maths Exercise 3.3<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"Choose_the_correct_answer_in_Exercises_11_and_12\"><\/span><strong>Choose the correct answer in Exercises 11 and 12.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>11. If A and B are symmetric matrices of same order, AB \u2013 BA is a:<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(A) Skew-symmetric matrix<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(B) Symmetric matrix<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(C) Zero matrix<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(S) Identity matrix<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: A and B are symmetric matrices <img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image136.png\" \/>\u00a0\u00a0A = A\u2019 and B = B\u2019<\/p>\n<p style=\"text-align: justify;\">Now, (AB \u2013 BA)\u2019 = (AB)\u2019 \u2013 (BA)\u2019\u00a0 <img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image136.png\" \/>\u00a0(AB \u2013 BA)\u2019 = B\u2019A\u2019 \u2013 A\u2019B\u2019 [Reversal law]\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image136.png\" \/>\u00a0(AB \u2013 BA)\u2019 = BA \u2013 AB [From eq. (i)] <img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image136.png\" \/>\u00a0(AB \u2013 BA)\u2019 = \u2013 (AB \u2013 BA)<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image004.png\" \/>\u00a0 (AB \u2013 BA) is a skew matrix.<\/p>\n<p style=\"text-align: justify;\">Therefore, option (A) is correct.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_33-8\"><\/span>NCERT Solutions class 12 Maths Exercise 3.3<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"12_If_A_then_A_A_I_if_the_value_of_is\"><\/span><strong>12. If A = <img decoding=\"async\" style=\"height: 49px; width: 121px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image137.png\" \/>, then A + A\u2019 = I, if the value of <img decoding=\"async\" style=\"height: 15px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image138.png\" \/>\u00a0is:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><strong>(A) <img decoding=\"async\" style=\"height: 42px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image139.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(B) <img decoding=\"async\" style=\"height: 42px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image140.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(C) <img decoding=\"async\" style=\"height: 15px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image141.png\" \/><\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>(D) <img decoding=\"async\" style=\"height: 42px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image142.png\" \/>\u00a0<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong>Ans. <\/strong>Given: A = <img decoding=\"async\" style=\"height: 49px; width: 116px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image143.png\" \/>\u00a0\u00a0Also A + A\u2019 = I<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image136.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 49px; width: 307px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image144.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image136.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 49px; width: 184px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image145.png\" \/><\/p>\n<p style=\"text-align: justify;\">Equating corresponding entries, we have<\/p>\n<p style=\"text-align: justify;\"><img decoding=\"async\" style=\"height: 19px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image146.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image136.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 42px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image147.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image136.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 42px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image148.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 15px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image136.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 42px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex3.3\/image149.png\" \/><\/p>\n<p style=\"text-align: justify;\">Therefore, option (B) is correct.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_33-9\"><\/span>NCERT Solutions class 12 Maths Exercise 3.3<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>NCERT Solutions Class 12 Maths PDF (Download) Free from myCBSEguide app and myCBSEguide website. Ncert solution class 12 Maths includes text book solutions from both part 1 and part 2. NCERT Solutions for CBSE Class 12 Maths have total 13 chapters. 12 Maths NCERT Solutions in PDF for free Download on our website. Ncert Maths class 12 solutions PDF and Maths ncert class 12 PDF solutions with latest modifications and as per the latest CBSE syllabus are only available in myCBSEguide<\/p>\n<h2><span class=\"ez-toc-section\" id=\"CBSE_App_for_Students\"><\/span>CBSE App for Students<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>To download\u00a0NCERT Solutions for class 12 Physics, Chemistry, Biology, History, Political Science, Economics, Geography, Computer Science, Home Science, Accountancy, Business Studies and Home Science; do check myCBSEguide app or website. myCBSEguide provides sample papers with solution, test papers for chapter-wise practice, NCERT solutions, NCERT Exemplar solutions, quick revision notes for ready reference, CBSE guess papers and CBSE important question papers. Sample Paper all are made available through\u00a0<a href=\"https:\/\/play.google.com\/store\/apps\/details?id=in.techchefs.MyCBSEGuide&amp;referrer=utm_source%3Dmycbse_bottom%26utm_medium%3Dtext%26utm_campaign%3Dmycbseads\"><strong>the best app for CBSE students<\/strong><\/a>\u00a0and myCBSEguide website.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>NCERT Solutions class 12 Maths Exercise 3.3 Class 12 Maths book solutions are available in PDF format for free download. These ncert book chapter wise questions and answers are very helpful for CBSE board exam. CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. Class 12 &#8230; <a title=\"NCERT Solutions class 12 Maths Exercise 3.3\" class=\"read-more\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-3-3\/\" aria-label=\"More on NCERT Solutions class 12 Maths Exercise 3.3\">Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":-1,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1432,281],"tags":[283,1342,321,216],"class_list":["post-4483","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematics-cbse-class-12","category-ncert-solutions","tag-cbse-study-material","tag-class-12","tag-mathematics","tag-ncert-solutions"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.0 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>NCERT Solutions class 12 Maths Exercise 3.3 | myCBSEguide<\/title>\n<meta name=\"description\" content=\"NCERT Solutions class 12 Maths Exercise 3.3 in PDF format for free download. 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