{"id":5065,"date":"2021-09-05T16:11:01","date_gmt":"2021-09-05T10:41:01","guid":{"rendered":"https:\/\/mathemerize.com\/?p=5065"},"modified":"2021-11-21T16:13:16","modified_gmt":"2021-11-21T10:43:16","slug":"minors-and-cofactors-of-a-matrix","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/minors-and-cofactors-of-a-matrix\/","title":{"rendered":"Minors and Cofactors of a Matrix (3×3 and 2×2) with Examples"},"content":{"rendered":"

Here you will learn how to find minors and cofactors of a matrix of order 3×3 and 2×2 with examples.<\/p>\n

Let’s begin –<\/p>\n

How to find Minors and Cofactors of a Matrix<\/h2>\n

Minor of Matrix (3×3 and 2×2)\u00a0<\/h3>\n

Let A = \\([a_{ij}]\\) be a square matrix of order n. The minor \\(M_{ij}\\) of \\(a_{ij}\\) in A is the determinant of the square sub-matrix of order (n – 1) obtained by leaving \\(i^{ith}\\) row and \\(j^{ith}\\) column of A.<\/p>\n

Example<\/span><\/strong> : if A = \\(\\begin{bmatrix} 4 & -7 \\\\ -3 & 2 \\end{bmatrix}\\), then<\/p>\n

\\(M_{11}\\) = Minor of\u00a0 \\(a_{11}\\) = 2,\u00a0 \\(M_{12}\\) = Minor of\u00a0 \\(a_{12}\\) = -3,\u00a0<\/p>\n

\\(M_{21}\\) = Minor of\u00a0 \\(a_{21}\\) = -7,\u00a0 \\(M_{22}\\) = Minor of\u00a0 \\(a_{22}\\) = 4<\/p>\n

If A = \\(\\begin{bmatrix} 1 & 2 & 3 \\\\ -3 & 2 & -1 \\\\ 2 & -4 & 3 \\end{bmatrix}\\), then<\/p>\n

\\(M_{11}\\) = Minor of\u00a0 \\(a_{11}\\)<\/p>\n

\\(\\implies\\) \\(M_{11}\\) = Determinant of the 2×2 square sub matrix obtained by leaving first row and first column of A<\/p>\n

\\(\\implies\\) \\(M_{11}\\) = \\(\\begin{vmatrix} 2 & -1 \\\\ -4 & 3 \\end{vmatrix}\\)<\/p>\n

Similarly, we obtain<\/p>\n

\\(M_{12}\\) = Minor of\u00a0 \\(a_{12}\\) = \\(\\begin{vmatrix} -3 & -1 \\\\\u00a0 2 & 3 \\end{vmatrix}\\) = -7<\/p>\n

\\(M_{13}\\) = Minor of\u00a0 \\(a_{13}\\) = \\(\\begin{vmatrix} -3 & 2 \\\\ 2 & -4 \\end{vmatrix}\\) = 8<\/p>\n

\\(M_{21}\\) = Minor of\u00a0 \\(a_{21}\\) = \\(\\begin{vmatrix} 2 & 3 \\\\ -4 & 3 \\end{vmatrix}\\) = 18<\/p>\n

\\(M_{22}\\) = Minor of\u00a0 \\(a_{22}\\) = \\(\\begin{vmatrix} 1 & 3 \\\\ 2 & 3 \\end{vmatrix}\\) = -3\u00a0 \u00a0 etc.<\/p>\n

Cofactor of Matrix (3×3 and 2×2)\u00a0<\/h3>\n

Let A = \\([a_{ij}]\\) be a square matrix of order n. The cofactor \\(C_{ij}\\) of \\(a_{ij}\\) in A is equal to \\((-1)^{i+j}\\) times the determinant of the sub-matrix of order (n – 1) obtained by leaving \\(i^{ith}\\) row and \\(j^{ith}\\) column of A.<\/p>\n

It follows from this definition that<\/p>\n

\\(C_{ij}\\) = Cofactor of \\(a_{ij}\\) in A = \\((-1)^{i+j}\\)\\(M_{ij}\\), where \\(M_{ij}\\) is minor of \\(a_{ij}\\) in A.<\/p>\n

Thus we have<\/p>\n

\n

\\(C_{ij}\\) = \\(M_{ij}\\) if i + j\u00a0 is even<\/p>\n

\\(C_{ij}\\) = \\(-M_{ij}\\) if i + j\u00a0 is odd<\/p>\n<\/blockquote>\n

Example<\/span><\/strong> : if A = \\(\\begin{bmatrix} 4 & -7 \\\\ -3 & 2 \\end{bmatrix}\\), then<\/p>\n

\\(C_{11}\\) = \\((-1)^{1+1}\\)\\(M_{11}\\) = \\(M_{11}\\)\u00a0 = 2, \\(C_{12}\\) = \\((-1)^{1+2}\\)\\(M_{12}\\) = -\\(M_{12}\\)\u00a0 = -(-3) = 3,<\/p>\n

\\(C_{21}\\) = \\((-1)^{2+1}\\)\\(M_{21}\\) = \\(M_{21}\\)\u00a0 = -(-7) = 7, \\(C_{22}\\) = \\((-1)^{2+2}\\)\\(M_{22}\\) = \\(M_{22}\\)\u00a0 = 4<\/p>\n

If A = \\(\\begin{bmatrix} 1 & 2 & 3 \\\\ -3 & 2 & -1 \\\\ 2 & -4 & 3 \\end{bmatrix}\\), then<\/p>\n

\\(C_{11}\\) = \\((-1)^{1+1}\\)\\(M_{11}\\) = \\(M_{11}\\) = \\(\\begin{vmatrix} 2 & -1 \\\\ -4 & 3 \\end{vmatrix}\\) = 2<\/p>\n

\\(C_{12}\\) = \\((-1)^{1+2}\\)\\(M_{12}\\) = -\\(M_{12}\\) = -\\(\\begin{vmatrix} -3 & -1 \\\\ 2 & 3 \\end{vmatrix}\\) = 7<\/p>\n

\\(C_{13}\\) = \\((-1)^{1+3}\\)\\(M_{13}\\) = \\(M_{13}\\) = \\(\\begin{vmatrix} -3 & 2 \\\\ 2 & -4 \\end{vmatrix}\\) = 8<\/p>\n

\\(C_{23}\\) = \\((-1)^{2+3}\\)\\(M_{23}\\) = -\\(M_{23}\\) = -\\(\\begin{vmatrix} 1 & 2 \\\\ 2 & -4 \\end{vmatrix}\\) = 8\u00a0 \u00a0 etc.<\/p>\n\n\n

\n
Previous – Properties of Determinant of Matrix<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn how to find minors and cofactors of a matrix of order 3×3 and 2×2 with examples. Let’s begin – How to find Minors and Cofactors of a Matrix Minor of Matrix (3×3 and 2×2)\u00a0 Let A = \\([a_{ij}]\\) be a square matrix of order n. The minor \\(M_{ij}\\) of \\(a_{ij}\\) in …<\/p>\n

Minors and Cofactors of a Matrix (3×3 and 2×2) with Examples<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[35],"tags":[218,211,216,217,215,214],"yoast_head":"\nMinors and Cofactors of a Matrix (3x3 and 2x2) with Examples<\/title>\n<meta name=\"description\" content=\"In this post you will learn how to find minors and cofactors of a matrix of order 3x3 and 2x2 with examples.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/minors-and-cofactors-of-a-matrix\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Minors and Cofactors of a Matrix (3x3 and 2x2) with Examples\" \/>\n<meta property=\"og:description\" content=\"In this post you will learn how to find minors and cofactors of a matrix of order 3x3 and 2x2 with examples.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathemerize.com\/minors-and-cofactors-of-a-matrix\/\" \/>\n<meta property=\"og:site_name\" content=\"Mathemerize\" \/>\n<meta property=\"article:published_time\" content=\"2021-09-05T10:41:01+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2021-11-21T10:43:16+00:00\" \/>\n<meta name=\"author\" content=\"mathemerize\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"mathemerize\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"3 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathemerize.com\/minors-and-cofactors-of-a-matrix\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/minors-and-cofactors-of-a-matrix\/\"},\"author\":{\"name\":\"mathemerize\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df\"},\"headline\":\"Minors and Cofactors of a Matrix (3×3 and 2×2) with Examples\",\"datePublished\":\"2021-09-05T10:41:01+00:00\",\"dateModified\":\"2021-11-21T10:43:16+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathemerize.com\/minors-and-cofactors-of-a-matrix\/\"},\"wordCount\":434,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathemerize.com\/#organization\"},\"keywords\":[\"cofactors\",\"matrix\",\"minor of a matrix\",\"minors\",\"minors and cofactors\",\"minors and cofactors of a matrix\"],\"articleSection\":[\"Determinants\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathemerize.com\/minors-and-cofactors-of-a-matrix\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathemerize.com\/minors-and-cofactors-of-a-matrix\/\",\"url\":\"https:\/\/mathemerize.com\/minors-and-cofactors-of-a-matrix\/\",\"name\":\"Minors and Cofactors of a Matrix (3x3 and 2x2) with Examples\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/#website\"},\"datePublished\":\"2021-09-05T10:41:01+00:00\",\"dateModified\":\"2021-11-21T10:43:16+00:00\",\"description\":\"In this post you will learn how to find minors and cofactors of a matrix of order 3x3 and 2x2 with examples.\",\"breadcrumb\":{\"@id\":\"https:\/\/mathemerize.com\/minors-and-cofactors-of-a-matrix\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mathemerize.com\/minors-and-cofactors-of-a-matrix\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mathemerize.com\/minors-and-cofactors-of-a-matrix\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/mathemerize.com\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Minors and Cofactors of a Matrix (3×3 and 2×2) with Examples\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/mathemerize.com\/#website\",\"url\":\"https:\/\/mathemerize.com\/\",\"name\":\"Mathemerize\",\"description\":\"Maths Tutorials - 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