{"id":1673,"date":"2021-05-26T12:07:46","date_gmt":"2021-05-26T06:37:46","guid":{"rendered":"https:\/\/www.goseeko.com\/blog\/?p=1673"},"modified":"2021-10-30T07:40:42","modified_gmt":"2021-10-30T07:40:42","slug":"what-is-cayeley-hamilton-theorem","status":"publish","type":"post","link":"https:\/\/www.goseeko.com\/blog\/what-is-cayeley-hamilton-theorem\/","title":{"rendered":"What is Cayeley-Hamilton Theorem?"},"content":{"rendered":"\n<h2 class=\"wp-block-heading\"><strong>Statement<\/strong><\/h2>\n\n\n\n<p>Cayeley-Hamilton theorem- <\/p>\n\n\n\n<p>According to Cayeley-Hamilton theorem, every square matrix satisfies its characteristic equation, that means for every square matrix of order n,<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/xHnuiO4-sDQhMR5_apX2f_pyOezojb600MuENxdcwWa5tFM237h1FwMGclO09cInRgA9Nrq-CAM7tXrmvSFLB2PAaZco86rdWHX_j9pg7TLpn_3WaqRmZyG1Fus7o0JaplnKwHuV\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<p>Then the matrix equation-<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/uDRrH-N1jbe92FEsRdidTwrqsn1ldKc2zo8hwouR5is5KuMpTt6Uj81svl24CJ1lvOXLYtg-dVqjTMcG4taPgA89e4E3IHGLViEkglbus7SV2VTt_H2W2hktduVvFIzrDtqoOpsl\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<p>Is satisfied by X = A<\/p>\n\n\n\n<p>That means<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/uR3zexh0aRgo3QL2acZONuOHd3mjuk8CidCezXgouNO3D-EIi-uhUuxftwLymZkBWGEB2q7Xjv1IWDdLgQ5rfqQ2-6IU1giBNymiDzSnJjDw-bnJpBpccZ1uxmHiLqYPXXOG8eJr\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<p><strong>Example-1: Find the characteristic equation of the matrix given below and Verify cayley-Hamlton theorem-<\/strong><\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/bWPuPTzwVN8_B__by31mBpG32Xf40Zyam5iwiyCyuSpeuOuQHJIi8RCz1hNpS8pXoBfLGPBnyqkJGj3Jsfk0ZsnK1M2htP-23_SSHgHD0hDlGfU-33V3W2PHGmH_HhIXY-A9fypL\" alt=\"\"\/><\/figure><\/div>\n\n\n\n<p>Sol. Characteristic equation of the matrix, we can be find as follows-<\/p>\n\n\n\n<figure class=\"wp-block-image is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/r5GxEdAwD6cKvCYPLztippMQuhYiohoVdLzLAnhognP1yAg3rOBOuIEpTVO0SBHeK4KTB7cq8aiAwQTfBt3MJp-oP9Irb-SMm-ImQD_RHLzlzZR4NWGZ-v5dbh__PhedoiCBVdZ3\" alt=\"\" width=\"159\" height=\"54\"\/><\/figure>\n\n\n\n<p>Which is-<\/p>\n\n\n\n<figure class=\"wp-block-image is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/mfayqFLS-qfTg1MYbgHwgqs7l_1qjeujmyHSFaj71MDuDYFmP-POOqdEm_nJWpk_q3hnGOZE4PN1REaHA2b77Bwi8_-y7gpyXLeB7Sil4rUxoWhclxkl9ImciQPpZ-2bFK52V3cC\" alt=\"\" width=\"454\" height=\"20\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/2DV4QZbykrds8DXupp1gcxyUtXwlklS6-SFVXMBAEPC5eDN1w_M9abr6LTi6uFI26NhnC5ZiD8sbvdKVfds4D0hXfOBSUy82nCQ4fs1HHOPi8lO3hV41mbpdUEsiMwZabi3v_Zam\" alt=\"\" width=\"158\" height=\"27\"\/><\/figure>\n\n\n\n<p>According to cayley-Hamilton theorem,<\/p>\n\n\n\n<figure class=\"wp-block-image is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/x9eaIz_p-afpecMhzh_noxpvKMjT4tz9BJ9K0hzFol7QKHLUmLD79OaXV_OFpebeok98ZraZBQmrh_E3ZvISRO3zCzdkX4518FND8g7yOXITFad2KcVpALcisPkEG8NEYA6rVvkE\" alt=\"\" width=\"261\" height=\"28\"\/><\/figure>\n\n\n\n<p>Now we will verify equation (1),<\/p>\n\n\n\n<figure class=\"wp-block-image is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/eQzsjAV3gbPySRFydtdh5mjNxpNsfbVj5ltEreU62AS-_zhjkZWGejjtNqjISb9YmYrr9F5nkfWqGSgvK8E7lekkeN39WD17x7xkykmrPbJkMyY6YoFxA2YBF3k3ZXqtCp_EggJT\" alt=\"\" width=\"400\" height=\"130\"\/><\/figure>\n\n\n\n<p>Put the required values in equation (1), we get<\/p>\n\n\n\n<figure class=\"wp-block-image is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/tzLdP7Bcf7fuqIE8igqWdZsJVkFUooWZMfpCSRBH8qVbQxLeoIwRTF6nQ_a28S16bjt7SE-yAwyE5PqyrV4ZC-3hrzXJ32XMFUcd0fGLPcB_dpLXlDhOKFS4ssGkF_UjxgRH2ZUB\" alt=\"\" width=\"441\" height=\"238\"\/><\/figure>\n\n\n\n<p>Hence the cayley-Hamilton theorem is verified.<\/p>\n\n\n\n<p><strong>Example: Find the characteristic equation of the the matrix A and verify Cayley-Hamilton theorem as well.<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/s2btiEqBd1rih2xZTPx133EcqpMvOZQx0DBuFC36EYEawJa92SQSq3ESuZSJVIQLHIOqgUxcmfaF5or390PhLSaV3_LvXEcvBNuxIl825pXQEzojqYZJPQGHhicaueGF1exNRYUf\" alt=\"\" width=\"148\" height=\"60\"\/><\/figure>\n\n\n\n<p>&nbsp;Sol. Characteristic equation will be-<\/p>\n\n\n\n<figure class=\"wp-block-image is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/lQ5DtiQbqEzNAMGzrfSGERmpeoupLu2Ok5J24LxZzztTkEoVJiJnu4fIn_JypOaLXsNKuIIpfosjyuWY2Jj4Il-35-7xWOtpDaA6pnkxqYG-JFJxjRyVXmmZUNvYTQrUre7FdeUO\" alt=\"\" width=\"580\" height=\"166\"\/><\/figure>\n\n\n\n<p>Which gives,<\/p>\n\n\n\n<figure class=\"wp-block-image is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/7YnS2Z8-72aagS6Tqr5fGgEOcF7Zf7urhGgpgvyW9WZDAAIPODYOVtnYWd2fRH6GTY7hsxQAcg-ofQ2QGS5uM5FctZEssQ0lrFm9UH_dvj8mY1mBBKdd7sR9GSXvC0L1wLIIaC8X\" alt=\"\" width=\"171\" height=\"28\"\/><\/figure>\n\n\n\n<p>Or<\/p>\n\n\n\n<figure class=\"wp-block-image is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/uzADk3vDvVHPc8ZXoAqkyEpZNvNzFUjxCAxmpnIuRinFjgLg0WspKBUn7zsHKaCMijW69moAcVJOvLxOK9Nxp-4y6J2nPi7X3gOdJup5nnAvVT-vyeM2NF4uUn2-sELMrYkXQkwp\" alt=\"\" width=\"160\" height=\"28\"\/><\/figure>\n\n\n\n<p>According to cayley-Hamilton theorem,<\/p>\n\n\n\n<figure class=\"wp-block-image is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/X1-qMClhp4s4mp165eFBfuvmSogVdBK4hdAkkGduELH5lYKIOOeKPe-qVQaEzLw8vgyiLnws8hv6Wz37a0suBR8ZYMy4Bq3ELpJnmtwEHI1eAO4gs5XMDIdKRAEf2mR_j8GgEOJU\" alt=\"\" width=\"336\" height=\"35\"\/><\/figure>\n\n\n\n<p>In order to verify cayley-Hamilton theorem , we will find the values of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/4DkIS6ezg0yBzg4ly-tEjb8PJlNcNFWhPzuYzk5ptmxATOci-NENH5fKfE4Qi26ESNZgC5Xq5J5lOAJaBBI2OlYsNkYep5YsQ0kNAHEJkv033ob8hwsaPelUTCDh2-w2bZgGJpXk\" width=\"24\" height=\"29\">and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/4i684UGfa3BIcAvBvRrgftG3oy-1KMs2YGOvP5jN3fMfP3nCq1iURuHLlSViQt7ADoD3DGqPogmh-qtxRk_uD0TrBRoM7nyaoEVEIZ793Bblx5bweuDR5LX7H5bVcnBcm7sXplu9\" width=\"19\" height=\"24\"><\/p>\n\n\n\n<p>So that,<\/p>\n\n\n\n<figure class=\"wp-block-image is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/bDuQIwaOzX7bXGmXcx_WKBNRA4QIH1IDmZbd8HZbyHmq0dk_GXDjji7IafrqE6L0ZZm9K-18EMXMBBdEiHY1OWwa8Uvr8OOl7OmQATOf4616nlPtXJNRTBmT76sCUseyDG8b_1vD\" alt=\"\" width=\"455\" height=\"65\"\/><\/figure>\n\n\n\n<p>Now<\/p>\n\n\n\n<figure class=\"wp-block-image is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/zHaYrHiig8lMWLg5jb8Nc0bo9NGINaoeFGGs7qVd5SJJiONYwC6G1FBxHvW7dYHraZjQwgX_Dv73ZmhsmxQEEamOMmgUnepUrYe9wKGDyP5BYGoH4PbhyPCGJIPpqpulxepAl4mK\" alt=\"\" width=\"464\" height=\"165\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/3urf4BcqszmGw8MDC5gcwVYshAyPJhlNCqAYDQwIRvZFlS3hZ15emZoaFtbTfkf90Xii-ypZNDtJdU_hbE5p9-rFkPLCqyNwf-NMU9GVKaXU5GZGC9eLkZjOmlQchrJcdZzuO0bB\" alt=\"\" width=\"185\" height=\"64\"\/><\/figure>\n\n\n\n<p>Put these values in equation(1), we get<\/p>\n\n\n\n<figure class=\"wp-block-image is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/JDvP2dZfYKkQYHjNO9RK_Gny57rzwIBbcpMLmkr15aQME23X_9UYx54148sVbvl1J5o1xmm-LwXQcVu9lVRFw97Gw-rCpWlQEr3dOt3lD9Eml1uK9L86U-aIWgvC5vZLESPtILv8\" alt=\"\" width=\"572\" height=\"233\"\/><\/figure>\n\n\n\n<p>Hence the cayley-hamilton theorem is verified.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Inverse of a matrix by Cayley-Hamilton theorem<\/strong><\/h2>\n\n\n\n<p>We can find the inverse of any matrix by multiplying the characteristic equation with A-1.<\/p>\n\n\n\n<p>For example, <\/p>\n\n\n\n<p>suppose we have a characteristic equation &nbsp;A<sup>3<\/sup>-5A<sup>2<\/sup>+7A-3I=0&nbsp; then multiply this by A-1 , then it becomes<\/p>\n\n\n\n<p class=\"has-text-align-center\">A<sup>2<\/sup>-5A+7I-3A<sup>-1<\/sup>=0<\/p>\n\n\n\n<p>Then we can find A<sup>-1<\/sup> by solving the above equation.<\/p>\n\n\n\n<p>Interested in learning about similar topics? Here are a few hand-picked blogs for you!<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li><a href=\"https:\/\/www.goseeko.com\/blog\/what-is-linear-programming\" target=\"_blank\" rel=\"noreferrer noopener\">What is linear programming?<\/a><\/li><li><a href=\"https:\/\/www.goseeko.com\/blog\/__trashed-4\" target=\"_blank\" rel=\"noreferrer noopener\">What is normal distribution?<\/a><\/li><li><a href=\"https:\/\/www.goseeko.com\/blog\/what-is-the-testing-of-hypothesis\" target=\"_blank\" rel=\"noreferrer noopener\">What is testing of hypothesis?<\/a><\/li><li><a href=\"https:\/\/www.goseeko.com\/blog\/what-is-conditional-probability\" target=\"_blank\" rel=\"noreferrer noopener\">What is conditional probability?<\/a><\/li><\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Every square matrix satisfies its characteristic equation, that means for every square matrix of order n,<\/p>\n","protected":false},"author":3,"featured_media":943,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[31],"tags":[],"class_list":["post-1673","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-maths"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v25.3.1 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>What is Cayeley-Hamilton Theorem? - Goseeko blog<\/title>\n<meta name=\"description\" content=\"According to Cayeley-Hamilton theorem, every square matrix satisfies its characteristic equation, that means for every square matrix of order n,\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.goseeko.com\/blog\/what-is-cayeley-hamilton-theorem\/\" \/>\n<meta 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