{"id":13433,"date":"2023-05-02T10:00:41","date_gmt":"2023-05-02T04:30:41","guid":{"rendered":"https:\/\/www.aplustopper.com\/?p=13433"},"modified":"2023-05-03T09:44:41","modified_gmt":"2023-05-03T04:14:41","slug":"multiply-matrices","status":"publish","type":"post","link":"https:\/\/www.aplustopper.com\/multiply-matrices\/","title":{"rendered":"How to Multiply Matrices"},"content":{"rendered":"<h2>How to Multiply Matrices<\/h2>\n<p><iframe loading=\"lazy\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/kuixY2bCc_0?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-128953\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/How-to-Multiply-Matrices-1.jpg\" alt=\"How to Multiply Matrices 1\" width=\"674\" height=\"411\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/How-to-Multiply-Matrices-1.jpg 674w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/How-to-Multiply-Matrices-1-300x183.jpg 300w\" sizes=\"(max-width: 674px) 100vw, 674px\" \/><\/p>\n<h3>Multiplication of matrices<\/h3>\n<p>Two matrices A and B are conformable for the product AB if the number of columns in A (pre-multiplier) is same as the number of rows in B (post multiplier). Thus, if A = [a<sub>ij<\/sub>]<sub>m\u00d7n \u00a0<\/sub>and B = [b<sub>ij<\/sub>]<sub>n\u00d7p <\/sub> are two matrices of order m \u00d7 n and n \u00d7\u00a0p respectively, then their product AB is of order and is defined as<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-128955\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/How-to-Multiply-Matrices-2.jpg\" alt=\"How to Multiply Matrices 2\" width=\"602\" height=\"350\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/How-to-Multiply-Matrices-2.jpg 602w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/How-to-Multiply-Matrices-2-300x174.jpg 300w\" sizes=\"(max-width: 602px) 100vw, 602px\" \/><br \/>\nThus, from (i), (AB)<sub>ij<\/sub>\u00a0= Sum of the product of elements of i<sup>th<\/sup> row of A with the corresponding elements of j<sup>th<\/sup> column of B.<\/p>\n<h3>Properties of matrix multiplication<\/h3>\n<p>If A,B and C are three matrices such that their product is defined, then<\/p>\n<ol>\n<li>AB \u2260 BA, (Generally not commutative)<\/li>\n<li>(AB)C = A(BC), (Associative Law)<\/li>\n<li>IA = A = AI, where I is identity matrix for matrix multiplication.<\/li>\n<li>A(B + C) = AB + AC, (Distributive law)<\/li>\n<li>If AB = AC \u21cf B = C, (Cancellation law is not applicable)<\/li>\n<li>If AB = 0, it does not mean that A = 0 or B = 0, again product of two non zero matrix may be a zero matrix.<\/li>\n<\/ol>\n<h3>Scalar multiplication of matrices<\/h3>\n<p>Let A = [a<sub>ij<\/sub>]<sub>m\u00d7n<\/sub>\u00a0be a matrix and k be a number, then the matrix which is obtained by multiplying every element of A by k is called scalar multiplication of A by k and it is denoted by kA.<br \/>\nThus, if A = [a<sub>ij<\/sub>]<sub>m\u00d7n<\/sub>, then kA = Ak = [ka<sub>ij<\/sub>]<sub>m\u00d7n<\/sub>.<\/p>\n<p><strong>Properties of scalar multiplication:<\/strong><br \/>\nIf A, B are matrices of the same order and \u03bb, \u03bc are any two scalars then<\/p>\n<ol>\n<li>\u03bb (A + B) = \u03bbA + \u03bbB<\/li>\n<li>(\u03bb + \u03bc)A = \u03bb + \u03bcA<\/li>\n<li>\u03bb(\u03bcA) = (\u03bb\u03bcA) = \u03bc(\u03bbA)<\/li>\n<li>(-\u03bbA) = -(\u03bbA) = \u03bb(-A)<\/li>\n<\/ol>\n<p>All the laws of ordinary algebra hold for the addition or subtraction of matrices and their multiplication by scalars.<\/p>\n<h3>Multiplication of Matrices Problems with Solutions<\/h3>\n<p><strong>1.<\/strong><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-128956\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/How-to-Multiply-Matrices-3.jpg\" alt=\"How to Multiply Matrices 3\" width=\"485\" height=\"303\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/How-to-Multiply-Matrices-3.jpg 485w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/How-to-Multiply-Matrices-3-300x187.jpg 300w\" sizes=\"(max-width: 485px) 100vw, 485px\" \/><br \/>\n<strong>Solution:<\/strong><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-128957\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/How-to-Multiply-Matrices-4.jpg\" alt=\"How to Multiply Matrices 4\" width=\"466\" height=\"99\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/How-to-Multiply-Matrices-4.jpg 466w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/How-to-Multiply-Matrices-4-300x64.jpg 300w\" sizes=\"(max-width: 466px) 100vw, 466px\" \/><br \/>\n<strong>2.<\/strong><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-128958\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/How-to-Multiply-Matrices-5.jpg\" alt=\"How to Multiply Matrices 5\" width=\"420\" height=\"211\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/How-to-Multiply-Matrices-5.jpg 420w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/How-to-Multiply-Matrices-5-300x151.jpg 300w\" sizes=\"(max-width: 420px) 100vw, 420px\" \/><br \/>\n<strong>Solution:<\/strong><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-128959\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/How-to-Multiply-Matrices-6.jpg\" alt=\"How to Multiply Matrices 6\" width=\"402\" height=\"207\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/How-to-Multiply-Matrices-6.jpg 402w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/How-to-Multiply-Matrices-6-300x154.jpg 300w\" sizes=\"(max-width: 402px) 100vw, 402px\" \/><br \/>\n<strong>3.<\/strong><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-128960\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/How-to-Multiply-Matrices-7.jpg\" alt=\"How to Multiply Matrices 7\" width=\"392\" height=\"125\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/How-to-Multiply-Matrices-7.jpg 392w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/How-to-Multiply-Matrices-7-300x96.jpg 300w\" sizes=\"(max-width: 392px) 100vw, 392px\" \/><br \/>\n<strong>Solution:<\/strong><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-128961\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/How-to-Multiply-Matrices-8.jpg\" alt=\"How to Multiply Matrices 8\" width=\"458\" height=\"293\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/How-to-Multiply-Matrices-8.jpg 458w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/How-to-Multiply-Matrices-8-300x192.jpg 300w\" sizes=\"(max-width: 458px) 100vw, 458px\" \/><br \/>\n<strong>4.<\/strong><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-128962\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/How-to-Multiply-Matrices-9.jpg\" alt=\"How to Multiply Matrices 9\" width=\"442\" height=\"120\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/How-to-Multiply-Matrices-9.jpg 442w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/How-to-Multiply-Matrices-9-300x81.jpg 300w\" sizes=\"(max-width: 442px) 100vw, 442px\" \/><br \/>\n<strong>Solution:<\/strong><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-128963\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/How-to-Multiply-Matrices-10.jpg\" alt=\"How to Multiply Matrices 10\" width=\"464\" height=\"318\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/How-to-Multiply-Matrices-10.jpg 464w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/How-to-Multiply-Matrices-10-300x206.jpg 300w\" sizes=\"(max-width: 464px) 100vw, 464px\" \/><\/p>\n","protected":false},"excerpt":{"rendered":"<p>How to Multiply Matrices Multiplication of matrices Two matrices A and B are conformable for the product AB if the number of columns in A (pre-multiplier) is same as the number of rows in B (post multiplier). Thus, if A = [aij]m\u00d7n \u00a0and B = [bij]n\u00d7p are two matrices of order m \u00d7 n and [&hellip;]<\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_genesis_hide_title":false,"_genesis_hide_breadcrumbs":false,"_genesis_hide_singular_image":false,"_genesis_hide_footer_widgets":false,"_genesis_custom_body_class":"","_genesis_custom_post_class":"","_genesis_layout":"","footnotes":""},"categories":[5],"tags":[4920,4924,4921,4923,4922],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v17.8 (Yoast SEO v17.8) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>How to Multiply Matrices - A Plus Topper<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.aplustopper.com\/multiply-matrices\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"How to Multiply Matrices\" \/>\n<meta property=\"og:description\" content=\"How to Multiply Matrices Multiplication of matrices Two matrices A and B are conformable for the product AB if the number of columns in A (pre-multiplier) is same as the number of rows in B (post multiplier). 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