{"id":5947,"date":"2021-10-04T17:15:56","date_gmt":"2021-10-04T11:45:56","guid":{"rendered":"https:\/\/mathemerize.com\/?p=5947"},"modified":"2022-03-06T14:15:54","modified_gmt":"2022-03-06T08:45:54","slug":"nilpotent-matrix","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/nilpotent-matrix\/","title":{"rendered":"Nilpotent Matrix – Definition and Example"},"content":{"rendered":"

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

Let’s begin –<\/p>\n

Nilpotent Matrix<\/h2>\n
\n

A square matrix of the order ‘n’ is said to be a nilpotent matrix of order m, m \\(\\in\\) N<\/p>\n

if \\(A^m\\) = O & \\(A^{m-1}\\) \\(\\ne\\) O.<\/p>\n<\/blockquote>\n

Example<\/span><\/strong> : Show that A = \\(\\begin{bmatrix} 1 & 1 &\u00a0 3 \\\\\u00a0 5 & 2 & 6 \\\\\u00a0 -2 & -1 & -3 \\end{bmatrix}\\) is a nilpotent matrix of order 3.<\/p>\n

Solution<\/span><\/strong> : We have given the matrix A,<\/p>\n

A = \\(\\begin{bmatrix} 1 & 1 &\u00a0 3 \\\\\u00a0 5 & 2 & 6 \\\\\u00a0 -2 & -1 & -3 \\end{bmatrix}\\)<\/p>\n

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

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

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

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

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

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

\\(\\therefore\\) \\(A^3\\) = O i.e. \\(A^k\\) = O<\/p>\n

Here k =3<\/p>\n

Hence A is a nilpotent matrix of order 3.<\/p>\n\n\n

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

Here you will learn what is nilpotent matrix with examples. Let’s begin – Nilpotent Matrix A square matrix of the order ‘n’ is said to be a nilpotent matrix of order m, m \\(\\in\\) N if \\(A^m\\) = O & \\(A^{m-1}\\) \\(\\ne\\) O. Example : Show that A = \\(\\begin{bmatrix} 1 & 1 &\u00a0 3 …<\/p>\n

Nilpotent 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":[464,465],"yoast_head":"\nNilpotent Matrix - Definition and Example - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn what is nilpotent 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\/nilpotent-matrix\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Nilpotent Matrix - 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