{"id":5954,"date":"2021-10-04T17:55:15","date_gmt":"2021-10-04T12:25:15","guid":{"rendered":"https:\/\/mathemerize.com\/?p=5954"},"modified":"2022-03-06T14:15:33","modified_gmt":"2022-03-06T08:45:33","slug":"involutory-matrix","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/involutory-matrix\/","title":{"rendered":"Involutory Matrix – Definition and Examples"},"content":{"rendered":"

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

Let’s begin –<\/p>\n

Involutory Matrix<\/h2>\n
\n

If \\(A^2\\) = I . the matrix A is said to be an involutory matrix, i.e. the square roots of the identity matrix (I) is involutory matrix.<\/p>\n

Note : The determinant value of this matrix (A) is 1 or -1.<\/p>\n<\/blockquote>\n

Example<\/span><\/strong> : Show that the matrix A = \\(\\begin{bmatrix} -5 & -8 &\u00a0 0 \\\\\u00a0 3 & 5 & 0 \\\\\u00a0 1 & 2 & -1 \\end{bmatrix}\\) is involutory.<\/p>\n

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

A = \\(\\begin{bmatrix} -5 & -8 &\u00a0 0 \\\\\u00a0 3 & 5 & 0 \\\\\u00a0 1 & 2 & -1 \\end{bmatrix}\\)<\/p>\n

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

\\(\\implies\\) \\(A^2\\) = \\(\\begin{bmatrix} -5 & -8 &\u00a0 0 \\\\\u00a0 3 & 5 & 0 \\\\\u00a0 1 & 2 & -1 \\end{bmatrix}\\) \\(\\times\\) \\(\\begin{bmatrix} -5 & -8 &\u00a0 0 \\\\\u00a0 3 & 5 & 0 \\\\\u00a0 1 & 2 & -1 \\end{bmatrix}\\)<\/p>\n

= \\(\\begin{bmatrix} 1 & 0 &\u00a0 0 \\\\\u00a0 0 & 1 & 0 \\\\\u00a0 0 & 0 & 1 \\end{bmatrix}\\)<\/p>\n

Hence, the given matrix A is involutary.<\/p>\n

Example<\/span><\/strong> : Show that the sqare matrix A is involutary, iff (I – A) (I + A) = O<\/p>\n

Solution<\/span><\/strong> : Let A is the involutary matrix,<\/p>\n

Then, \\(A^2\\) = I<\/p>\n

(I – A) (I + A) = \\(I^2\\) + IA – AI – \\(A^2\\)<\/p>\n

= I + A – A – \\(A^2\\)\u00a0<\/p>\n

= I – \\(A^2\\) = O<\/p>\n

Conversely, let (I – A) (I + A) = O<\/p>\n

\\(\\implies\\) \\(I^2\\) + IA – AI – \\(A^2\\) = O<\/p>\n

\\(\\implies\\) I + A – A – \\(A^2\\) = O<\/p>\n

= I – \\(A^2\\) = O<\/p>\n

Hence, the given matrix A is involutary.<\/p>\n\n\n

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

Here you will learn what is involutory matrix with examples. Let’s begin – Involutory Matrix If \\(A^2\\) = I . the matrix A is said to be an involutory matrix, i.e. the square roots of the identity matrix (I) is involutory matrix. Note : The determinant value of this matrix (A) is 1 or -1. …<\/p>\n

Involutory Matrix – Definition and 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":[34],"tags":[462,463],"yoast_head":"\nInvolutory Matrix - Definition and Examples - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn what is involutory 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\/involutory-matrix\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Involutory Matrix - 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