{"id":5943,"date":"2021-10-04T16:24:19","date_gmt":"2021-10-04T10:54:19","guid":{"rendered":"https:\/\/mathemerize.com\/?p=5943"},"modified":"2022-03-06T14:13:17","modified_gmt":"2022-03-06T08:43:17","slug":"periodic-matrix","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/periodic-matrix\/","title":{"rendered":"Periodic Matrix – Definition and Example"},"content":{"rendered":"

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

Let’s begin –<\/p>\n

Periodic Matrix<\/h2>\n
\n

A square matrix which satisfies the relation \\(A^{k+1}\\) = A for some positive integer k, is called a periodic matrix.<\/p>\n

The period of the matrix is the least value of k for which \\(A^{k+1}\\) = A holds true.<\/p>\n

Note that the period of idempotent matrix is 1.<\/p>\n<\/blockquote>\n

Example<\/span><\/strong> : Find the period of the matrix A = \\(\\begin{bmatrix} 1 & -2 &\u00a0 -6 \\\\\u00a0 -3 & 2 & 9 \\\\\u00a0 2 & 0 & -3 \\end{bmatrix}\\).<\/p>\n

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

A = \\(\\begin{bmatrix} 1 & -2 &\u00a0 -6 \\\\\u00a0 -3 & 2 & 9 \\\\\u00a0 2 & 0 & -3 \\end{bmatrix}\\).<\/p>\n

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

\\(\\implies\\) \\(A^2\\) = \\(\\begin{bmatrix} 1 & -2 &\u00a0 -6 \\\\\u00a0 -3 & 2 & 9 \\\\\u00a0 2 & 0 & 3 \\end{bmatrix}\\) \\(\\times\\) \\(\\begin{bmatrix} 1 & -2 &\u00a0 -6 \\\\\u00a0 -3 & 2 & 9 \\\\\u00a0 2 & 0 & 3 \\end{bmatrix}\\)<\/p>\n

= \\(\\begin{bmatrix} 5 & -6 &\u00a0 -6 \\\\\u00a0 9 & 10 & 9 \\\\\u00a0 -4 & -4 & -3 \\end{bmatrix}\\).<\/p>\n

Now, \\(A^3\\) = \\(A^2\\).A<\/p>\n

\\(\\implies\\) \\(A^3\\) = \\(\\begin{bmatrix} 5 & -6 &\u00a0 -6 \\\\\u00a0 9 & 10 & 9 \\\\\u00a0 -4 & -4 & -3 \\end{bmatrix}\\) \\(\\times\\) \\(\\begin{bmatrix} 1 & -2 &\u00a0 -6 \\\\\u00a0 -3 & 2 & 9 \\\\\u00a0 2 & 0 & -3 \\end{bmatrix}\\)<\/p>\n

= \\(\\begin{bmatrix} 1 & -2 &\u00a0 -6 \\\\\u00a0 -3 & 2 & 9 \\\\\u00a0 2 & 0 & -3 \\end{bmatrix}\\) = A<\/p>\n

Hence, \\(A^3\\) = A. comparing it with the equation \\(A^{k+1}\\) = A gives k = 2.<\/p>\n

So, Period of the given matrix is 2.<\/p>\n\n\n

\n
Next – Nilpotent Matrix \u2013 Definition and Example<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Idempotent Matrix – Definition and Example<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn what is periodic matrix with examples. Let’s begin – Periodic Matrix A square matrix which satisfies the relation \\(A^{k+1}\\) = A for some positive integer k, is called a periodic matrix. The period of the matrix is the least value of k for which \\(A^{k+1}\\) = A holds true. Note that …<\/p>\n

Periodic 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":[466,467],"yoast_head":"\nPeriodic Matrix - Definition and Example - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn what is periodic 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\/periodic-matrix\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Periodic Matrix - 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