{"id":5937,"date":"2021-10-04T15:49:14","date_gmt":"2021-10-04T10:19:14","guid":{"rendered":"https:\/\/mathemerize.com\/?p=5937"},"modified":"2021-11-26T20:32:45","modified_gmt":"2021-11-26T15:02:45","slug":"idempotent-matrix","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/idempotent-matrix\/","title":{"rendered":"Idempotent Matrix – Definition and Example"},"content":{"rendered":"

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

Let’s begin –<\/p>\n

Idempotent Matrix<\/h2>\n
\n

A square matrix is idempotent matrix provided \\(A^2\\) = A.<\/p>\n

For this matrix note the following :<\/p>\n

(i) \\(A^n\\) = A \\(\\forall\\) n \\(\\ge\\) 2, n \\(\\in\\) N.<\/p>\n

(ii) The determinant value of this matrix is either 1 or 0.<\/p>\n<\/blockquote>\n

Example<\/span><\/strong> : Show that the matrix A = \\(\\begin{bmatrix} 2 & -2 &\u00a0 -4 \\\\\u00a0 -1 & 3 & 4 \\\\\u00a0 1 & -2 & -3 \\end{bmatrix}\\) is idempotent.<\/p>\n

Solution<\/span><\/strong> : We have,<\/p>\n

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

Now, \\(A^2\\) = A.A<\/p>\n

\\(\\implies\\) A = \\(\\begin{bmatrix} 2 & -2 &\u00a0 -4 \\\\\u00a0 -1 & 3 & 4 \\\\\u00a0 1 & -2 & -3 \\end{bmatrix}\\) \\(\\times\\) \\(\\begin{bmatrix} 2 & -2 &\u00a0 -4 \\\\\u00a0 -1 & 3 & 4 \\\\\u00a0 1 & -2 & -3 \\end{bmatrix}\\)<\/p>\n

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

Hence, matrix A is idempotent.<\/p>\n

Example<\/span><\/strong> : Find the determinant of above matrix A = \\(\\begin{bmatrix} 2 & -2 &\u00a0 -4 \\\\\u00a0 -1 & 3 & 4 \\\\\u00a0 1 & -2 & -3 \\end{bmatrix}\\)<\/p>\n

Solution<\/span><\/strong> : We have,<\/p>\n

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

Now, | A | = \\(\\begin{vmatrix} 2 & -2 &\u00a0 -4 \\\\\u00a0 -1 & 3 & 4 \\\\\u00a0 1 & -2 & -3 \\end{vmatrix}\\)<\/p>\n

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

\\(\\implies\\) | A | = 2 (-9 + 8) + 2 (3 – 4) – 4 ( 2 – 3)\u00a0<\/p>\n

= 2(-1) + 2(-1) – 4(-1)<\/p>\n

= -2 –\u00a0 2 + 4 = 0<\/p>\n

Hence, determinant of matrix A is 0.<\/p>\n\n\n

\n
Next – Periodic Matrix – Definition and Example<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn what is idempotent matrix with examples. Let’s begin – Idempotent Matrix A square matrix is idempotent matrix provided \\(A^2\\) = A. For this matrix note the following : (i) \\(A^n\\) = A \\(\\forall\\) n \\(\\ge\\) 2, n \\(\\in\\) N. (ii) The determinant value of this matrix is either 1 or 0. …<\/p>\n

Idempotent Matrix – Definition and 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":[468,469],"yoast_head":"\nIdempotent Matrix - Definition and Example - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn what is idempotent matrix 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\/idempotent-matrix\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Idempotent Matrix - 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