{"id":69005,"date":"2023-09-09T12:02:00","date_gmt":"2023-09-09T06:32:00","guid":{"rendered":"https:\/\/www.kopykitab.com\/blog\/?p=69005"},"modified":"2023-12-01T10:39:47","modified_gmt":"2023-12-01T05:09:47","slug":"rd-sharma-solutions-class-12-maths-chapter-5-exercise-5-3","status":"publish","type":"post","link":"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-12-maths-chapter-5-exercise-5-3\/","title":{"rendered":"RD Sharma Class 12 Solutions Chapter 5 Algebra of Matrices Exercise 5.3 (Updated for 2024)"},"content":{"rendered":"\n<p><img class=\"alignnone size-full wp-image-119603\" src=\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/01\/RD-Sharma-Solutions-Class-12-Chapter-5-Exercise-5.3.png\" alt=\"RD Sharma Solutions Class 12 Maths Chapter 5 Exercise 5.3\" width=\"1200\" height=\"675\" srcset=\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/01\/RD-Sharma-Solutions-Class-12-Chapter-5-Exercise-5.3.png 1200w, https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/01\/RD-Sharma-Solutions-Class-12-Chapter-5-Exercise-5.3-768x432.png 768w\" sizes=\"(max-width: 1200px) 100vw, 1200px\" \/><\/p>\n<p><strong>RD Sharma Solutions Class 12 Maths Chapter 5 Exercise 5.3:\u00a0<\/strong>RD Sharma Solutions for Class 12 Ex 5.3 are available in PDF format on the Kopykitab website, with solutions carefully compiled by an expert team. It is primarily intended for students to utilize as study material in order to pass the Class 12 exams with a decent grade in accordance with the CBSE syllabus. The fifth chapter&#8217;s Exercise 5.3 presents questions that are solved using the transpose of a matrix. The <a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions\/\"><strong>RD Sharma Solutions<\/strong><\/a> are written in an explanatory style to help students understand the concepts. <a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-class-12-solutions\/\"><strong>RD Sharma Solutions for Class 12 Maths<\/strong><\/a> Chapter 5 Algebra of Matrices Exercise 5.3 are provided in this article.<\/p>\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_47_1 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\">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=\"ez-toc-toggle-icon-1\"><label for=\"item-69e7628a34d1d\" aria-label=\"Table of Content\"><span style=\"display: flex;align-items: center;width: 35px;height: 30px;justify-content: center;direction:ltr;\"><svg 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ez-toc-heading-7\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-12-maths-chapter-5-exercise-5-3\/#what-is-the-rd-sharma-solutions-class-12-maths-chapter-5-exercise-53\" title=\"What is the RD Sharma Solutions Class 12 Maths Chapter 5 Exercise 5.3?\">What is the RD Sharma Solutions Class 12 Maths Chapter 5 Exercise 5.3?<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"download-rd-sharma-solutions-class-12-maths-chapter-5-exercise-53-free-pdf\"><\/span>Download RD Sharma Solutions Class 12 Maths Chapter 5 Exercise 5.3 Free PDF<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<div id=\"example1\" style=\"text-align: justify;\">\u00a0<\/div>\n<p style=\"text-align: justify;\"><style>\n.pdfobject-container { height: 800px;}<br \/>\n.pdfobject { border: 1px solid #666; }<br \/>\n<\/style><\/p>\n<p style=\"text-align: justify;\"><script src=\"https:\/\/www.kopykitab.com\/_utility\/js\/pdfobject.min.js\"><\/script><br \/><script>PDFObject.embed(\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/01\/RD-_Sharma-_Class_12_Chapter_05_5.3.pdf\", \"#example1\");<\/script><\/p>\n<p><a href=\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/01\/RD-_Sharma-_Class_12_Chapter_05_5.3.pdf\">RD Sharma Solutions Class 12 Maths Chapter 5 Exercise 5.3<\/a><\/p>\n<h3><span class=\"ez-toc-section\" id=\"access-answers-to-rd-sharma-solutions-for-chapter-5-%e2%80%93-algebra-of-matrices-ex-53-class-12-maths-important-questions-with-solution\"><\/span>Access answers to RD Sharma Solutions For Chapter 5 \u2013 Algebra of Matrices Ex 5.3 Class 12 Maths Important Questions With Solution<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>1. Compute the indicated products:<\/strong><\/p>\n<p><strong><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-60.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 203\" \/><\/strong><\/p>\n<p><strong><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-61.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 204\" \/><\/strong><\/p>\n<p><strong><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-62.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 205\" \/><\/strong><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>(i) Consider<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-141.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 206\" \/><\/p>\n<p>On simplification, we get,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-142.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 207\" \/><\/p>\n<p>(ii) Consider<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-143.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 208\" \/><\/p>\n<p>On simplification, we get,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-144.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 209\" \/><\/p>\n<p>(iii) Consider<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-145.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 210\" \/><\/p>\n<p>On simplification, we get,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-146.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 211\" \/><\/p>\n<p><strong>2. Show that AB \u2260 BA in each of the following cases:<\/strong><\/p>\n<p><strong><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-63.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 212\" \/><\/strong><\/p>\n<p><strong><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-64.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 213\" \/><\/strong><\/p>\n<p><strong><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-65.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 214\" \/><\/strong><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>(i) Consider,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-147.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 215\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-148.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 216\" \/><\/p>\n<p>Again consider,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-149.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 217\" \/><\/p>\n<p>From equations (1) and (2), it is clear that<\/p>\n<p>AB \u2260 BA<\/p>\n<p>(ii) Consider,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-150.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 218\" \/><\/p>\n<p>Now again, consider,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-151.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 219\" \/><\/p>\n<p>From equations (1) and (2), it is clear that<\/p>\n<p>AB \u2260 BA<\/p>\n<p>(iii) Consider,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-152.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 220\" \/><\/p>\n<p>Now again, consider,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-153.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 221\" \/><\/p>\n<p>From equations (1) and (2), it is clear that<\/p>\n<p>AB \u2260 BA<\/p>\n<p><strong>3. Compute the products AB and BA, whichever exists in each of the following cases:<\/strong><\/p>\n<p><strong><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-66.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 222\" \/><\/strong><\/p>\n<p><strong><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-67.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 223\" \/><\/strong><\/p>\n<p><strong><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-68.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 224\" \/><\/strong><\/p>\n<p><strong><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-69.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 225\" \/><\/strong><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>(i) Consider,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-154.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 226\" \/><\/p>\n<p>BA does not exist.<\/p>\n<p>Because the number of columns in B is greater than the rows in A.<\/p>\n<p>(ii) Consider,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-155.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 227\" \/><\/p>\n<p>Again consider,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-156.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 228\" \/><\/p>\n<p>(iii) Consider,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-157.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 229\" \/><\/p>\n<p>AB = [0 + (-1) + 6 + 6]<\/p>\n<p>AB = 11<\/p>\n<p>Again consider,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-158.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 230\" \/><\/p>\n<p>(iv) Consider,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-70.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 231\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-159.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 232\" \/><\/p>\n<p><strong>4. Show that AB \u2260 BA in each of the following cases:<\/strong><\/p>\n<p><strong><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-71.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 233\" \/><\/strong><\/p>\n<p><strong><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-72.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 234\" \/><\/strong><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>(i) Consider,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-160.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 235\" \/><\/p>\n<p>Again consider,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-161.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 236\" \/><\/p>\n<p>From equations (1) and (2), it is clear that<\/p>\n<p>AB \u2260 BA<\/p>\n<p>(ii) Consider,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-162.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 237\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-163.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 238\" \/><\/p>\n<p>Again consider,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-164.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 239\" \/><\/p>\n<p>From equations (1) and (2), it is clear that,<\/p>\n<p>AB \u2260 BA<\/p>\n<p><strong>5. Evaluate the following:<\/strong><\/p>\n<p><strong><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-165.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 240\" \/><\/strong><\/p>\n<p><strong><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-73.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 241\" \/><\/strong><\/p>\n<p><strong><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-166.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 242\" \/><\/strong><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>(i) Given<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-167.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 243\" \/><\/p>\n<p>First, we have to add the first two matrices.<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-168.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 244\" \/><\/p>\n<p>On simplifying, we get<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-169.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 245\" \/><\/p>\n<p>(ii) Given,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-74.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 246\" \/><\/p>\n<p>First, we have to multiply the first two given matrices.<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-170.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 247\" \/><\/p>\n<p>= 82<\/p>\n<p>(iii) Given<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-171.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 248\" \/><\/p>\n<p>First, we have to subtract the matrix which is inside the bracket.<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-172.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 249\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-173.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 250\" \/><\/p>\n<p><strong><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-75.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 251\" \/><\/strong><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-174.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 252\" \/><\/p>\n<p>We know that,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-175.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 253\" \/><\/p>\n<p>Again we know that,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-176.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 254\" \/><\/p>\n<p>Now, consider,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-177.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 255\" \/><\/p>\n<p>We have,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-178.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 256\" \/><\/p>\n<p>Now, from equations (1), (2), (3) and (4), it is clear that A<sup>2\u00a0<\/sup>= B<sup>2<\/sup>= C<sup>2<\/sup>= I<sub>2<\/sub><\/p>\n<p><strong><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-76.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 257\" \/><\/strong><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-77.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 258\" \/><\/p>\n<p>Consider,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-179.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 259\" \/><\/p>\n<p>Now we have to find,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-180.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 260\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-181.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 261\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-78.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 262\" \/><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/c-users-tnluser-downloads-codecogseqn-74-gif.gif\" alt=\"C:\\Users\\tnluser\\Downloads\\CodeCogsEqn (74).gif\" \/><\/p>\n<p>Consider,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-182.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 264\" \/><\/p>\n<p>Hence the proof.<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-79.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 265\" \/><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-80.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 266\" \/><\/p>\n<p>Consider,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-183.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 267\" \/><\/p>\n<p>Again consider,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-184.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 268\" \/><\/p>\n<p>Hence the proof.<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-81.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 269\" \/><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-82.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 270\" \/><\/p>\n<p>Consider,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-185.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 271\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-186.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 272\" \/><\/p>\n<p>Hence the proof.<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-83.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 273\" \/><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-84.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 274\" \/><\/p>\n<p>Consider,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-187.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 275\" \/><\/p>\n<p>We know that,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-188.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 276\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-189.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 277\" \/><\/p>\n<p>Again we have,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-190.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 278\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-191.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 279\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-85.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 280\" \/><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-86.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 281\" \/><\/p>\n<p>Consider,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-192.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 282\" \/><\/p>\n<p>Again consider,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-193.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 283\" \/><\/p>\n<p>From equations (1) and (2), AB = BA = 0<sub>3\u00d73<\/sub><\/p>\n<p><strong><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-87.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 284\" \/><\/strong><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-88.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 285\" \/><\/p>\n<p>Consider,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-194.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 286\" \/><\/p>\n<p>Again consider,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-195.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 287\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-196.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 288\" \/><\/p>\n<p>From equations (1) and (2), AB = BA = 0<sub>3\u00d73<\/sub><\/p>\n<p><strong><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-89.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 289\" \/><\/strong><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-90.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 290\" \/><\/p>\n<p>Now consider,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-197.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 291\" \/><\/p>\n<p>Therefore AB = A<\/p>\n<p>Again consider BA we get,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-198.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 292\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-199.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 293\" \/><\/p>\n<p>Hence, BA = B<\/p>\n<p>Hence, the proof.<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-91.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 294\" \/><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-92.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 295\" \/><\/p>\n<p>Consider,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-200.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 296\" \/><\/p>\n<p>Now again, consider, B<sup>2<\/sup><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-201.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 297\" \/><\/p>\n<p>Now by subtracting equation (2) from equation (1) we get,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-202.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 298\" \/><\/p>\n<p><strong>16. For the following matrices, verify the associativity of matrix multiplication, i.e., (AB) C = A (BC)<\/strong><\/p>\n<p><strong><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-93.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 299\" \/><\/strong><\/p>\n<p><strong><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-94.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 300\" \/><\/strong><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>(i) Given<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-95.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 301\" \/><\/p>\n<p>Consider,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-203.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 302\" \/><\/p>\n<p>Now consider RHS,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-204.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 303\" \/><\/p>\n<p>From equations (1) and (2), it is clear that (AB) C = A (BC)<\/p>\n<p>(ii) Given,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-96.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 304\" \/><\/p>\n<p>Consider the LHS,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-205.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 305\" \/><\/p>\n<p>Now consider RHS,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-206.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 306\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-207.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 307\" \/><\/p>\n<p>From equations (1) and (2), it is clear that (AB) C = A (BC)<\/p>\n<p><strong>17. For the following matrices, verify the distributivity of matrix multiplication over matrix addition, i.e., A (B + C) = AB + AC.<\/strong><\/p>\n<p><strong><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-97.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 308\" \/><\/strong><\/p>\n<p><strong><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-98.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 309\" \/><\/strong><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>(i) Given<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-99.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 310\" \/><\/p>\n<p>Consider LHS,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-208.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 311\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-209.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 312\" \/><\/p>\n<p>Now consider RHS,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-210.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 313\" \/><\/p>\n<p>From equations (1) and (2), it is clear that A (B + C) = AB + AC<\/p>\n<p>(ii) Given,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-100.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 314\" \/><\/p>\n<p>Consider the LHS<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-211.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 315\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-212.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 316\" \/><\/p>\n<p>Now consider RHS,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-213.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 317\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-214.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 318\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-215.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 319\" \/><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-216.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 320\" \/><\/p>\n<p>Consider the LHS,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-217.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 321\" \/><\/p>\n<p>Now consider RHS<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-218.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 322\" \/><\/p>\n<p>From the above equations, LHS = RHS<\/p>\n<p>Therefore, A (B \u2013 C) = AB \u2013 AC.<\/p>\n<p><strong>19. Compute the elements a<sub>43<\/sub>\u00a0and a<sub>22<\/sub>\u00a0of the matrix:<\/strong><\/p>\n<p><strong><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-101.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 323\" \/><\/strong><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-102.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 324\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-219.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 325\" \/><\/p>\n<p>From the above matrix, a<sub>43<\/sub>\u00a0= 8and a<sub>22<\/sub>\u00a0= 0<\/p>\n<p><strong><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-103.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 326\" \/><\/strong><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-220.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 327\" \/><\/p>\n<p>Consider,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-221.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 328\" \/><\/p>\n<p>Again consider,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-222.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 329\" \/><\/p>\n<p>Now, consider the RHS<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-223.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 330\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-224.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 331\" \/><\/p>\n<p>Therefore, A<sup>3<\/sup>\u00a0= p I + q A + rA<sup>2<\/sup><\/p>\n<p>Hence the proof.<\/p>\n<p><strong>21. If \u03c9 is a complex cube root of unity, show that<\/strong><\/p>\n<p><strong><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-104.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 332\" \/><\/strong><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-105.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 333\" \/><\/p>\n<p>It is also given that \u03c9 is a complex cube root of unity,<\/p>\n<p>Consider the LHS,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-106.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 334\" \/><\/p>\n<p>We know that 1 + \u03c9 + \u03c9<sup>2<\/sup>\u00a0= 0 and \u03c9<sup>3<\/sup>\u00a0= 1<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-107.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 335\" \/><\/p>\n<p>Now by simplifying we get,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-108.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 336\" \/><\/p>\n<p>Again by substituting 1 + \u03c9 + \u03c9<sup>2<\/sup>\u00a0= 0 and \u03c9<sup>3<\/sup>\u00a0= 1 in above matrix we get,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-109.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 337\" \/><\/p>\n<p>Therefore LHS = RHS<\/p>\n<p>Hence the proof.<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-110.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 338\" \/><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-111.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 339\" \/><\/p>\n<p>Consider A<sup>2<\/sup><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-225.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 340\" \/><\/p>\n<p>Therefore A<sup>2<\/sup>\u00a0= A<\/p>\n<p><strong><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-112.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 341\" \/><\/strong><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-113.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 342\" \/><\/p>\n<p>Consider A<sup>2<\/sup>,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-226.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 343\" \/><\/p>\n<p>Hence A<sup>2<\/sup>\u00a0= I<sub>3<\/sub><\/p>\n<p><strong><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-114.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 344\" \/><\/strong><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>(i) Given<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-115.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 345\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-227.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 346\" \/><\/p>\n<p>= [2x + 1 + 2 + x + 3] = 0<\/p>\n<p>= [3x + 6] = 0<\/p>\n<p>= 3x = -6<\/p>\n<p>x = -6\/3<\/p>\n<p>x = -2<\/p>\n<p>(ii) Given,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-116.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 347\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-228.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 348\" \/><\/p>\n<p>On comparing the above matrix, we get,<\/p>\n<p>x = 13<\/p>\n<p><strong><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-117.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 349\" \/><\/strong><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-118.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 350\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-229.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 351\" \/><\/p>\n<p>\u21d2\u00a0[(2x + 4) x + 4 (x + 2) \u2013 1(2x + 4)] = 0<\/p>\n<p>\u21d2\u00a02x<sup>2<\/sup>\u00a0+ 4x + 4x + 8 \u2013 2x \u2013 4 = 0<\/p>\n<p>\u21d2\u00a02x<sup>2<\/sup>\u00a0+ 6x + 4 = 0<\/p>\n<p>\u21d2\u00a02x<sup>2<\/sup>\u00a0+ 2x + 4x + 4 = 0<\/p>\n<p>\u21d2\u00a02x (x + 1) + 4 (x + 1) = 0<\/p>\n<p>\u21d2\u00a0(x + 1) (2x + 4) = 0<\/p>\n<p>\u21d2\u00a0x = -1 or x = -2<\/p>\n<p>Hence, x = -1 or x = -2<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-119.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 352\" \/><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-230.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 353\" \/><\/p>\n<p>By multiplying, we get,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-231.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 354\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-232.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 355\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-120.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 356\" \/><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-121.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 357\" \/><\/p>\n<p>Now we have to prove A<sup>2<\/sup>\u00a0\u2013 A + 2 I = 0<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-233.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 358\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-234.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 359\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-122.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 360\" \/><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-123.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 361\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-235.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 362\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-236.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 363\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-237.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 364\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-124.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 365\" \/><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-125.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 366\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-238.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 367\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-239.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 368\" \/><\/p>\n<p>Hence the proof.<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-126.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 369\" \/><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-127.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 370\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-240.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 371\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-241.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 372\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-242.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 373\" \/><\/p>\n<p>Hence the proof.<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-128.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 374\" \/><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-129.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 375\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-243.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 376\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-244.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 377\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-245.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 378\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-130.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 379\" \/><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-131.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 380\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-246.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 381\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-247.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 382\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-132.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 383\" \/><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-133.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 384\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-248.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 385\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-249.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 386\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-250.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 387\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-134.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 388\" \/><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-135.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 389\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-251.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 390\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-252.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 391\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-253.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 392\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-254.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 393\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-255.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 394\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-136.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 395\" \/><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-137.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 396\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-256.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 397\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-257.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 398\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-258.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 399\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-138.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 400\" \/><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-139.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 401\" \/><\/p>\n<p>I is the identity matrix, so<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-259.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 402\" \/><\/p>\n<p>Also given,<br \/><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-260.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 403\" \/><\/p>\n<p>Now, we have to find A<sup>2<\/sup>, we get<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-261.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 404\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-262.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 405\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-263.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 406\" \/><\/p>\n<p>Now, we will find the matrix for 8A, and we get<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-264.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 407\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-265.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 408\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-266.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 409\" \/><\/p>\n<p>So,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-267.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 410\" \/><\/p>\n<p>Substitute corresponding values from eqns (i) and (ii), we get<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-268.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 411\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-269.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 412\" \/><\/p>\n<p>And to satisfy the above condition of equality, the corresponding entries of the matrices should be equal<\/p>\n<p>Hence,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-270.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 413\" \/><\/p>\n<p>Therefore, the value of k is 7<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-140.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 414\" \/><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-141.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 415\" \/><\/p>\n<p>To show that\u00a0f (A) = 0<\/p>\n<p>Substitute\u00a0x = A\u00a0in\u00a0f(x), we get<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-271.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 416\" \/><\/p>\n<p>I is the identity matrix, so<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-272.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 417\" \/><\/p>\n<p>Now, we will find the matrix for A<sup>2<\/sup>, and we get<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-273.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 418\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-274.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 419\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-275.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 420\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-276.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 421\" \/><\/p>\n<p>Now, we will find the matrix for 2A, and we get<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-277.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 422\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-278.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 423\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-279.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 424\" \/><\/p>\n<p>Substitute corresponding values from eqns (ii) and (iii) in eqn (i), we get<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-280.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 424\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-281.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 426\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-282.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 427\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-283.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 428\" \/><\/p>\n<p>So,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-284.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 429\" \/><\/p>\n<p>Hence Proved<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-142.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 430\" \/><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-143.gif\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 431\" \/><\/p>\n<p>So<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-285.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 432\" \/><\/p>\n<p>Now, we will find the matrix for A<sup>2<\/sup>, and we get<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-286.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 433\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-287.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 434\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-288.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 435\" \/><\/p>\n<p>Now, we will find the matrix for \u03bb A, and we get<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-289.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 436\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-290.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 437\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-291.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 438\" \/><\/p>\n<p>But given, A<sup>2<\/sup>\u00a0= \u03bb A + \u03bc I<\/p>\n<p>Substitute corresponding values from equations (i) and (ii), we get<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-292.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 439\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-293.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 440\" \/><\/p>\n<p>And to satisfy the above condition of equality, the corresponding entries of the matrices should be equal<\/p>\n<p>Hence, \u03bb + 0 = 4\u00a0\u21d2\u00a0\u03bb = 4<\/p>\n<p>And also, 2\u03bb + \u03bc = 7<\/p>\n<p>Substituting the obtained value of \u03bb in the above equation, we get<\/p>\n<p>2(4) + \u03bc = 7\u00a0\u21d2\u00a08 + \u03bc = 7\u00a0\u21d2\u00a0\u03bc = \u2013 1<\/p>\n<p>Therefore, the values of \u03bb and \u03bc are 4 and \u2013 1, respectively<\/p>\n<p><strong>39. Find the value of x for which the matrix product<\/strong><\/p>\n<p><strong><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-294.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 441\" \/><\/strong><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>We know,<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-295.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 442\" \/><\/p>\n<p>is the identity matrix of size 3.<\/p>\n<p>So, according to the given criteria<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-296.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 443\" \/><\/p>\n<p>Now we will multiply the two matrices on LHS using the formula c<sub>ij<\/sub>\u00a0= a<sub>i1<\/sub>b<sub>1j<\/sub>\u00a0+ a<sub>i2<\/sub>b<sub>2j<\/sub>\u00a0+ \u2026 + a<sub>in\u00a0<\/sub>b<sub>nj<\/sub>, we get<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-297.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 444\" \/><\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-298.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 445\" \/><\/p>\n<p>And to satisfy the above condition of equality, the corresponding entries of the matrices should be equal<\/p>\n<p>So we get<\/p>\n<p><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-299.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 446\" \/><\/p>\n<p>So the value of x is<br \/><img title=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/12\/rd-sharma-solutions-for-class-12-maths-chapter-5-i-300.png\" alt=\"RD Sharma Solutions for Class 12 Maths Chapter 5 Image 447\" \/><\/p>\n<h3><span class=\"ez-toc-section\" id=\"access-other-exercises-of-rd-sharma-solutions-for-class-12-chapter-5\"><\/span>Access other exercises of RD Sharma Solutions For Class 12 Chapter 5\u00a0<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<ul>\n<li><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-12-maths-chapter-5-exercise-5-1\/\">RD Sharma Class 12 Maths Exercise 5.1 Solutions<\/a><\/li>\n<li><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-12-maths-chapter-5-exercise-5-2\/\">RD Sharma Class 12 Maths Exercise 5.2 Solutions<\/a><\/li>\n<li><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-12-maths-chapter-5-exercise-5-4\/\">RD Sharma Class 12 Maths Exercise 5.4 Solutions<\/a><\/li>\n<li><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-12-maths-chapter-5-exercise-5-5\/\">RD Sharma Class 12 Maths Exercise 5.5 Solutions<\/a><\/li>\n<\/ul>\n<p>We have provided complete details of\u00a0<a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-12-maths-chapter-5\/\"><strong>RD Sharma Solutions Class 12 Maths Chapter 5<\/strong><\/a> Exercise 5.3. If you have any queries related to <b>the CBSE Class 12 Exam, please<\/b>\u00a0ask us in the comment section below.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"faqs-on-rd-sharma-solutions-class-12-maths-chapter-5-exercise-53\"><\/span>FAQs on RD Sharma Solutions Class 12 Maths Chapter 5 Exercise 5.3<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n<div id=\"rank-math-faq\" class=\"rank-math-block\">\n<div class=\"rank-math-list \">\n<div id=\"faq-question-1629901232731\" class=\"rank-math-list-item\">\n<h3 class=\"rank-math-question \"><span class=\"ez-toc-section\" id=\"where-can-i-get-rd-sharma-solutions-class-12-maths-chapter-5-exercise-53-free-pdf\"><\/span>Where can I get RD Sharma Solutions Class 12 Maths Chapter 5 Exercise 5.3 Free PDF?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<div class=\"rank-math-answer \">\n\n<p>You can get RD Sharma Solutions Class 12 Maths Chapter 5 Exercise 5.3 Free PDF from the above article.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"faq-question-1629901434338\" class=\"rank-math-list-item\">\n<h3 class=\"rank-math-question \"><span class=\"ez-toc-section\" id=\"list-out-the-benefits-of-using-kopykitab%e2%80%99s-rd-sharma-solutions-for-class-12-maths-chapter-5-exercise-53\"><\/span>List out the benefits of using Kopykitab\u2019s RD Sharma Solutions for Class 12 Maths Chapter 5 Exercise 5.3.<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<div class=\"rank-math-answer \">\n\n<p>1. Expert subject teachers at Kopykitab design the solutions in order to improve speed and accuracy.<br \/>2. Students can easily get solutions in an exercise-wise and chapter-wise format.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"faq-question-1683098181514\" class=\"rank-math-list-item\">\n<h3 class=\"rank-math-question \"><span class=\"ez-toc-section\" id=\"what-is-the-rd-sharma-solutions-class-12-maths-chapter-5-exercise-53\"><\/span>What is the RD Sharma Solutions Class 12 Maths Chapter 5 Exercise 5.3?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<div class=\"rank-math-answer \">\n\n<p>The RD Sharma Solutions Class 12 Maths Chapter 5 Exercise 5.3 in the blog above.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>RD Sharma Solutions Class 12 Maths Chapter 5 Exercise 5.3:\u00a0RD Sharma Solutions for Class 12 Ex 5.3 are available in PDF format on the Kopykitab website, with solutions carefully compiled by an expert team. It is primarily intended for students to utilize as study material in order to pass the Class 12 exams with a &#8230; <a title=\"RD Sharma Class 12 Solutions Chapter 5 Algebra of Matrices Exercise 5.3 (Updated for 2024)\" class=\"read-more\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-12-maths-chapter-5-exercise-5-3\/\" aria-label=\"More on RD Sharma Class 12 Solutions Chapter 5 Algebra of Matrices Exercise 5.3 (Updated for 2024)\">Read more<\/a><\/p>\n","protected":false},"author":238,"featured_media":119603,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"fifu_image_url":"","fifu_image_alt":""},"categories":[73411,2985,73410],"tags":[3429,73223,73664,4388],"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/posts\/69005"}],"collection":[{"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/users\/238"}],"replies":[{"embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/comments?post=69005"}],"version-history":[{"count":5,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/posts\/69005\/revisions"}],"predecessor-version":[{"id":515142,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/posts\/69005\/revisions\/515142"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/media\/119603"}],"wp:attachment":[{"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/media?parent=69005"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/categories?post=69005"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/tags?post=69005"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}