{"id":10017,"date":"2022-02-09T15:56:54","date_gmt":"2022-02-09T10:26:54","guid":{"rendered":"https:\/\/mathemerize.com\/?p=10017"},"modified":"2022-02-09T15:56:58","modified_gmt":"2022-02-09T10:26:58","slug":"matrix-polynomial-definition-and-example","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/matrix-polynomial-definition-and-example\/","title":{"rendered":"Matrix Polynomial – Definition and Example"},"content":{"rendered":"

Here you will learn what is the matrix polynomial definition with examples.<\/p>\n

Let’s begin –<\/p>\n

Matrix Polynomial<\/h2>\n
\n

Definition<\/strong> : Let f(x) = \\(a_0x^n\\) + \\(a_1x^{n – 1}\\) + \\(a_2x^{n – 2}\\) + ……. + \\(a_{n – 1}x\\) + \\(a_n\\) be a polynomial and let A be a square matrix of order n. Then,<\/p>\n

f(A) = \\(a_0A^n\\) + \\(a_1A^{n – 1}\\) + \\(a_2A^{n – 2}\\) + ….\u00a0 \\(a_{n – 1}A\\) + \\(a_nI^n\\)<\/p>\n

is called a matrix polynomial.<\/p>\n<\/blockquote>\n

For example, if f(x) = \\(x^2\\) – 3x + 2 is a polynomial and A is a square matrix, then f(A) = \\(A^2\\) – 3A + 2I is a matrix polynomial.<\/p>\n

Also Read<\/strong> : Different Types of Matrices \u2013 Definitions and Examples<\/a><\/p>\n

Example<\/strong><\/span> : Let f(x) =\u00a0 \\(x^2\\) – 4x + 7. Find f(A), if A = \\(\\begin{bmatrix} 2 & 3 \\\\ -1 &\u00a0 2 \\end{bmatrix}\\).<\/p>\n

Solution<\/span><\/strong> : We have, f(x) =\u00a0 \\(x^2\\) – 4x + 7<\/p>\n

\\(\\therefore\\)\u00a0 \u00a0f(A) = \\(A^2 – 4A + 7I_2\\)<\/p>\n

Now,\u00a0 \\(A^2\\) = \\(\\begin{bmatrix} 2 & 3 \\\\ -1 &\u00a0 2 \\end{bmatrix}\\)\\(\\begin{bmatrix} 2 & 3 \\\\ -1 &\u00a0 2 \\end{bmatrix}\\)<\/p>\n

= \\(\\begin{bmatrix} 4 – 3 & 6 + 6 \\\\ -2 – 2 &\u00a0 -3 + 4 \\end{bmatrix}\\) = \\(\\begin{bmatrix} 1 & 12 \\\\ -4 &\u00a0 1 \\end{bmatrix}\\)<\/p>\n

-4A = \\(\\begin{bmatrix} -8 & -12 \\\\ 4 &\u00a0 -8 \\end{bmatrix}\\)<\/p>\n

and, \\(7I_2\\) = \\(\\begin{bmatrix} 7 & 0 \\\\ 0 &\u00a0 7 \\end{bmatrix}\\)<\/p>\n

\\(\\therefore\\)\u00a0 \u00a0f(A) = \\(A^2 – 4A + 7I_2\\)<\/p>\n

\\(\\implies\\)\u00a0 f(A) = \\(\\begin{bmatrix} 1 & 12 \\\\ -4 &\u00a0 1 \\end{bmatrix}\\) + \\(\\begin{bmatrix} -8 & -12 \\\\ 4 &\u00a0 -8 \\end{bmatrix}\\) + \\(\\begin{bmatrix} 7 & 0 \\\\ 0 &\u00a0 7 \\end{bmatrix}\\)<\/p>\n

\\(\\implies\\)\u00a0 f(A) = \\(\\begin{bmatrix} 0 & 0 \\\\ 0 &\u00a0 0 \\end{bmatrix}\\)<\/p>\n\n\n

<\/p>\n","protected":false},"excerpt":{"rendered":"

Here you will learn what is the matrix polynomial definition with examples. Let’s begin – Matrix Polynomial Definition : Let f(x) = \\(a_0x^n\\) + \\(a_1x^{n – 1}\\) + \\(a_2x^{n – 2}\\) + ……. + \\(a_{n – 1}x\\) + \\(a_n\\) be a polynomial and let A be a square matrix of order n. Then, f(A) = …<\/p>\n

Matrix Polynomial – Definition and Example<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","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":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[34],"tags":[884,885],"yoast_head":"\nMatrix Polynomial - Definition and Example<\/title>\n<meta name=\"description\" content=\"In this post you will learn what is the matrix polynomial definition with examples based on it.\" \/>\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\/matrix-polynomial-definition-and-example\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Matrix Polynomial - 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