{"id":5001,"date":"2021-09-04T00:22:28","date_gmt":"2021-09-03T18:52:28","guid":{"rendered":"https:\/\/mathemerize.com\/?p=5001"},"modified":"2021-11-26T20:38:09","modified_gmt":"2021-11-26T15:08:09","slug":"formula-for-inverse-of-a-matrix","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/formula-for-inverse-of-a-matrix\/","title":{"rendered":"Formula for Inverse of a Matrix – Properties, Example"},"content":{"rendered":"

Here you will learn formula for inverse of a matrix and properties of inverse of matrix with example.<\/p>\n

Let’s begin –<\/p>\n

Formula for Inverse of a Matrix<\/h2>\n

A square matrix A said to be invertible if and only if it is non-singular (i.e. |A| \\(\\ne\\) 0) and there exists a matrix B such that, AB = I = BA.<\/p>\n

B is called the inverse (reciprocal) of A and is denoted by \\(A^{-1}\\). Thus<\/p>\n

\\(A^{-1}\\) = B \\(\\iff\\) AB = I = BA<\/p>\n

We have, A.(adj A) = | A | \\(I_n\\)<\/p>\n

\\(A^{-1}\\).A(adj A) = \\(A^{-1}\\) \\(I_n\\) | A |\u00a0<\/p>\n

\\(I_n\\) (adj A) = \\(A^{-1}\\) | A | \\(I_n\\)<\/p>\n

\\(\\therefore\\) \\(A^{-1}\\) = \\((adj A)\\over | A |\\)<\/p>\n

\n

Inverse of matrix A is \\(A^{-1}\\) = \\((adj A)\\over | A |\\)<\/p>\n

Note<\/strong> : The necessary and sufficient condition for a square matrix A to be invertible is that | A | \\(\\ne\\) 0<\/p>\n<\/blockquote>\n

Also Read<\/strong> : How to find Adjoint of the Matrix (2\u00d72 & 3\u00d73)<\/a><\/p>\n

Example<\/span><\/strong> : find the inverse of the matrix \\(\\begin{bmatrix} 2 & -1\u00a0 \\\\ 3 & 4\u00a0 \\end{bmatrix}\\)<\/p>\n

Solution<\/span> <\/strong>: Let A = \\(\\begin{bmatrix} 2 & -1\u00a0 \\\\ 3 & 4\u00a0 \\end{bmatrix}\\). Then,<\/p>\n

| A | = \\(\\begin{vmatrix} 2 & -1\u00a0 \\\\ 3 & 4\u00a0 \\end{vmatrix}\\) = 8 + 3 = 11 \\(\\ne\\) 0<\/p>\n

So, A is a non-singular matrix ( i.e. | A | \\(\\ne\\) 0 ) and therefore it is invertible. Let \\(C_{ij}\\) be cofactor of \\(a_{ij}\\) in A.Then the cofactors of elements of A are given by<\/p>\n

\\(C_{11}\\) = 4, \\(C_{12}\\) = -3, \\(C_{21}\\) = -(-1) = 1 and \\(C_{22}\\) = 2.<\/p>\n

\\(\\therefore\\) adj A = \\({\\begin{bmatrix} 4 & -3\u00a0 \\\\ 1 & 2\u00a0 \\end{bmatrix}}^T\\) = \\(\\begin{bmatrix} 4 & 1\u00a0 \\\\ -3 & 2\u00a0 \\end{bmatrix}\\)<\/p>\n

Hence, \\(A^{-1}\\) = \\(1\\over | A |\\) adj A = \\(1\\over 11\\) \\(\\begin{bmatrix} 4 & 1\u00a0 \\\\ -3 & 2\u00a0 \\end{bmatrix}\\)<\/p>\n

Properties of Inverse\u00a0<\/h2>\n

(1) (Cancellation Law) Let A, B, C be square matrices of the same order n. If A is a non singular matrix, then<\/p>\n

(i) AB = AC \\(\\implies\\) B = C<\/p>\n

(ii) BA = CA \\(\\implies\\) B = C<\/p>\n

(2) (Reversal Law) If A and B are invertible matrices of the same order, then AB is invertible and \\((AB)^{-1}\\) = \\(B^{-1}\\)\\(A^{-1}\\).<\/p>\n

(3) If A is invertible square matrix, then \\(A^T\\) is also invertible and \\((A^T)^{-1}\\) = \\((A^{-1})^T\\).<\/p>\n

(4) The invertible of an invertible symmetric matrix is a symmetric matrix.<\/p>\n

(5) Let A be a non singular square matrix of order n. Then, | adj A | = \\(| A |^{n-1}\\).<\/p>\n

(6) If A and B are non singular square matrices of the same order, then adj AB = (adj A)(adj B)<\/p>\n

(7) If A is an invertible square matrix, then \\(adj A^T\\) = \\((adj A)^T\\).<\/p>\n

(8) The adjoint of a symmetric matrix is also a symmetric matrix.<\/p>\n

(9) If A is a non singular square matrix, then adj(adj A) = \\(| A |^{n-2}\\) A.<\/p>\n

Note :\u00a0\u00a0<\/strong>If A is a non singular matrix of order n, then |adj(adj A)| = \\(|A|^{(n-1)^2}\\).<\/p>\n

(10) If the product of non-null square matrices is a null matrix, then both of them must be a singular matrices.<\/p>\n

(11) If A is a non singular matrix, then \\(|A^{-1}|\\) = \\(|A|^{n-1}\\) i.e. \\(|A^{-1}|\\) = \\(1\\over |A|\\).<\/p>\n\n\n

\n
Previous – Adjoint of the Matrix (2×2 & 3×3) – Properties, Examples<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn formula for inverse of a matrix and properties of inverse of matrix with example. Let’s begin – Formula for Inverse of a Matrix A square matrix A said to be invertible if and only if it is non-singular (i.e. |A| \\(\\ne\\) 0) and there exists a matrix B such that, AB …<\/p>\n

Formula for Inverse of a Matrix – Properties, Example<\/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":[34],"tags":[470,471,472],"yoast_head":"\nFormula for Inverse of a Matrix - Properties, Example - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn formula for inverse of a matrix and properties of inverse of matrix with example.\" \/>\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\/formula-for-inverse-of-a-matrix\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Formula for Inverse of a Matrix - 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