{"id":617841,"date":"2023-02-27T09:26:14","date_gmt":"2023-02-27T09:26:14","guid":{"rendered":"https:\/\/www.indcareer.com\/schools\/?p=617841"},"modified":"2024-05-02T10:18:05","modified_gmt":"2024-05-02T10:18:05","slug":"ncert-exemplar-class-12-maths-chapter-3-matrices","status":"publish","type":"post","link":"https:\/\/www.indcareer.com\/schools\/ncert-exemplar-class-12-maths-chapter-3-matrices\/","title":{"rendered":"NCERT Exemplar Class 12 Maths Chapter 3: Matrices"},"content":{"rendered":"\n<p>NCERT Exemplar Class 12 Maths Chapter 3: Matrices. NCERT Exemplar Solutions for Class 12 Maths Chapter 3 Matrices prepare students for their Class 12 exams thoroughly. <\/p>\n\n\n\n<p>Maths problems and solutions for the Class 12 pdf are provided here which are similar to the questions being asked in the previous year&#8217;s board. <\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-ncert-exemplar-class-12-maths-chapter-3-matrices\"><strong>NCERT Exemplar Class 12 Maths Chapter 3: Matrices<\/strong><\/h2>\n\n\n\n<p><\/p>\n\n\n\n<p><strong>3.1 Overview<\/strong><br><strong>3.1.1&nbsp;<\/strong>A matrix is an ordered rectangular array of numbers (or functions). For example,<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49059115927_cf673eb71c_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<p>The numbers (or functions) are called the elements or the entries of the matrix. The horizontal lines of elements are said to constitute rows of the matrix and the vertical lines of elements are said to constitute columns of the matrix.<\/p>\n\n\n\n<p><strong>3.1.2 Order of a Matrix<\/strong><\/p>\n\n\n\n<p>A matrix having m rows and n columns is called a matrix of order m \u00d7 n or simply m \u00d7 n matrix (read as an m by n matrix).In the above example, we have A as a matrix of order 3 \u00d7 3 i.e., 3 \u00d7 3 matrix.<br>In general, an m \u00d7 n matrix has the following rectangular array :<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49058899816_ebb1f0ce08_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<p>The element, a<sub>ij<\/sub>&nbsp;is an element lying in the i<sup>th<\/sup>&nbsp;row and j<sup>th<\/sup>&nbsp;column and is known as the (i, j)<sup>th<\/sup>&nbsp;element of A. The number of elements in an m \u00d7 n matrix will be equal to mn.<\/p>\n\n\n\n<p><strong>3.1.3 Types of Matrices<\/strong><\/p>\n\n\n\n<p>(i) A matrix is said to be a row matrix if it has only one row.<br>(ii) A matrix is said to be a column matrix if it has only one column.<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49058418358_97b486bc60_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<p><strong>3.1.4 Additon of Matrices<\/strong><\/p>\n\n\n\n<p>Two matrices can be added if they are of the same order.<\/p>\n\n\n\n<p><strong>3.1.5 Multiplication of Matrix by a Scalar<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49058939086_e82fd1e169_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<p><strong>3.1.6 Negative of a Matrix<\/strong><\/p>\n\n\n\n<p>The negative of a matrix A is denoted by \u2013A. We define \u2013A = (\u20131)A.<\/p>\n\n\n\n<p><strong>3.1.7 Multiplication of Matrices<\/strong><\/p>\n\n\n\n<p>The multiplication of two matrices A and B is defined if the number of columns of A is equal to the number of rows of B.<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49058435643_f50a43b2b0_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<p><strong>Notes:<\/strong><br>1. If AB is defined, then BA need not be defined.<br>2. If A, B are, respectively m \u00d7 n, k \u00d7 l matrices, then both AB and BA are defined if and only if n = k and l = m.<br>3. If AB and BA are both defined, it is not necessary that AB = BA.<br>4. If the product of two matrices is a zero matrix, it is not necessary that one of the matrices is a zero matrix.<br>5. For three matrices A, B and C of the same order, if A = B, then AC = BC, but converse is not true.<br>6. A. A = A<sup>2<\/sup>&nbsp;, A. A. A = A<sup>3<\/sup>, so on.<\/p>\n\n\n\n<p><strong>3.1.8 Transpose of a Matrix<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49058448958_b0f8d42f3d_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<p><strong>3.1.9 Symmetric Matrix and Skew Symmetric Matrix<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49058454603_b85b8b0fe3_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49058454528_5c5dd8d2e8_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<p><strong>Note<\/strong>&nbsp;: Diagonal elements of a skew symmetric matrix are zero.<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49058971591_204770d109_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<p><strong>3.1.10 Invertible Matrices<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49059198962_d1612913ca_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<p><strong>3.1.11 Inverse of a Matrix using Elementary Row or Column Operations<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49058473873_1091736232_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">Short Answer Type Questions (Solved Examples)<\/h3>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49062792311_ba92803549_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49062279008_1ac3832ecd_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49062792256_1e317e3f53_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">Long Answer Type Questions (Solved Examples)<\/h3>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49063020162_ea663c82af_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49062288248_beb8ca70b2_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49063020007_1763473acb_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49062801506_8514785b85_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49062801471_4b604b60ac_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49062288023_a538bea3f5_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49063019832_4db8a21c85_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">Multiple Choice Questions (Solved Examples)<\/h3>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49063028407_22b51f0b4f_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49062810246_7f73b8b514_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49062810166_fd6152b769_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">Fill in the Blanks Type Questions (Solved Examples)<\/h3>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49062303483_9efffe487b_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">True or False Statements Type Questions (Solved Examples)<\/h3>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49062817506_247e587f4c_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">Short Answer Type Questions (Exercise)<\/h3>\n\n\n\n<p>Click Here&nbsp;to download NCERT Exemplar Class 12 Maths Unit 3 Matrices.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Long Answer Type Questions (Exercise)<\/h3>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/15173037423_48690d7a37_o.jpg\" alt=\"Class 12 Important Questions for Maths -\u00a0Matrices\"\/><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">Multiple Choice Questions (Exercise)<\/h3>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/15794035232_bf17d0baf0_o.jpg\" alt=\"Class 12 Important Questions for Maths -\u00a0Matrices\"\/><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">Fill In Blanks Type Questions (Exercise)<\/h3>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/15173037353_9d05bca47e_o.jpg\" alt=\"Class 12 Important Questions for Maths -\u00a0Matrices\"\/><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">True or False Statements Type Questions (Exercise)<\/h3>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/15790597441_c68a32aa5c_o.jpg\" alt=\"Class 12 Important Questions for Maths -\u00a0Matrices\"\/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">Answers<\/h2>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49063051872_9307aea011_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49062834331_0d3917aa28_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2024\/05\/49062834266_55ebeb3115_o.png\" alt=\"\"\/><\/figure>\n\n\n\n<p>Class 12: Maths Chapter 3 solutions. Complete Class 12 Maths Chapter 3 Notes.<\/p>\n\n\n\n<div class=\"wp-block-buttons is-layout-flex wp-block-buttons-is-layout-flex\">\n<div class=\"wp-block-button\"><a class=\"wp-block-button__link has-background wp-element-button\" href=\"https:\/\/www.indcareer.com\/schools\/ncert-exemplar-problems-solutions\/\" style=\"background-color:#f06d6d\" target=\"_blank\" rel=\"noreferrer noopener\">NCERT Exemplar Solutions<\/a><\/div>\n\n\n\n<div class=\"wp-block-button\"><a class=\"wp-block-button__link has-background wp-element-button\" href=\"https:\/\/www.indcareer.com\/schools\/ncert-exemplar-class-12-Maths-important-questions\/\" style=\"background-color:#f06d6d\" target=\"_blank\" rel=\"noreferrer noopener\">NCERT Class 12 Exemplar Solutions<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>NCERT Exemplar Class 12 Maths Chapter 3: Matrices. NCERT Exemplar Solutions for Class 12 Maths Chapter 3 Matrices prepare students for their Class 12 exams thoroughly. Maths problems and solutions for the Class 12 pdf are provided here which are similar to the questions being asked in the previous year&#8217;s board. NCERT Exemplar Class 12 [&hellip;]<\/p>\n","protected":false},"author":302,"featured_media":617883,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"newspack_featured_image_position":"","newspack_post_subtitle":"","newspack_article_summary_title":"Overview:","newspack_article_summary":"","newspack_hide_updated_date":false,"newspack_show_updated_date":false,"footnotes":""},"categories":[1411,25],"tags":[],"boards":[],"class_list":["post-617841","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-book-solutions","category-class-12","entry"],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v27.0 (Yoast SEO v27.1.1) - https:\/\/yoast.com\/product\/yoast-seo-premium-wordpress\/ -->\n<title>NCERT Exemplar for Class 12, Maths Chapter 3 - IndCareer Schools<\/title>\n<meta name=\"description\" content=\"NCERT Exemplar Class 12 Maths Chapter 3: Matrices | Browse all Class 12 Maths Chapters NCERT Exemplar books - IndCareer Schools\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.indcareer.com\/schools\/ncert-exemplar-class-12-maths-chapter-3-matrices\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"NCERT Exemplar Class 12 Maths Chapter 3: Matrices\" \/>\n<meta property=\"og:description\" content=\"NCERT Exemplar Class 12 Maths Chapter 3: Matrices. NCERT Exemplar Solutions for Class 12 Maths Chapter 3 Matrices prepare students for their Class 12\" \/>\n<meta property=\"og:url\" content=\"https:\/\/www.indcareer.com\/schools\/ncert-exemplar-class-12-maths-chapter-3-matrices\/\" \/>\n<meta property=\"og:site_name\" content=\"IndCareer Schools\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/indcareer\" \/>\n<meta property=\"article:published_time\" content=\"2023-02-27T09:26:14+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-05-02T10:18:05+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2023\/02\/indcareer-schools-2-60-6-scaled.jpg\" \/>\n\t<meta property=\"og:image:width\" content=\"1600\" \/>\n\t<meta property=\"og:image:height\" content=\"901\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/jpeg\" \/>\n<meta name=\"author\" content=\"Pooja\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:creator\" content=\"@indcareer\" \/>\n<meta name=\"twitter:site\" content=\"@indcareer\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Pooja\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"10 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/www.indcareer.com\/schools\/ncert-exemplar-class-12-maths-chapter-3-matrices\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/www.indcareer.com\/schools\/ncert-exemplar-class-12-maths-chapter-3-matrices\/\"},\"author\":{\"name\":\"Pooja\",\"@id\":\"https:\/\/www.indcareer.com\/schools\/#\/schema\/person\/d6945cf059726f162259ba738092301e\"},\"headline\":\"NCERT Exemplar Class 12 Maths Chapter 3: Matrices\",\"datePublished\":\"2023-02-27T09:26:14+00:00\",\"dateModified\":\"2024-05-02T10:18:05+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/www.indcareer.com\/schools\/ncert-exemplar-class-12-maths-chapter-3-matrices\/\"},\"wordCount\":545,\"publisher\":{\"@id\":\"https:\/\/www.indcareer.com\/schools\/#organization\"},\"image\":{\"@id\":\"https:\/\/www.indcareer.com\/schools\/ncert-exemplar-class-12-maths-chapter-3-matrices\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2023\/02\/indcareer-schools-2-60-6-scaled.jpg\",\"articleSection\":[\"Book Solutions\",\"Class 12\"],\"inLanguage\":\"en-US\"},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/www.indcareer.com\/schools\/ncert-exemplar-class-12-maths-chapter-3-matrices\/\",\"url\":\"https:\/\/www.indcareer.com\/schools\/ncert-exemplar-class-12-maths-chapter-3-matrices\/\",\"name\":\"NCERT Exemplar for Class 12, Maths Chapter 3 - 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