{"id":4911,"date":"2021-08-31T17:54:07","date_gmt":"2021-08-31T17:54:07","guid":{"rendered":"https:\/\/mathemerize.com\/?p=4911"},"modified":"2021-11-26T20:57:59","modified_gmt":"2021-11-26T15:27:59","slug":"properties-of-multiplication-of-matrices","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/properties-of-multiplication-of-matrices\/","title":{"rendered":"Properties of Multiplication of Matrices"},"content":{"rendered":"

Here you will learn properties of multiplication of matrices, positive integral powers of square matrix and matrix polynomial.<\/p>\n

Let’s begin –<\/p>\n

Also Read : Multiplication of Matrices \u2013 Examples & Definition<\/a><\/p>\n

Properties of Multiplication of Matrices<\/h2>\n

(a) Matrix multiplication is not commutative in general i.e AB \\(\\ne\\) BA.<\/p>\n

(b) Matrix multiplication is associative i.e. (AB) C = A (BC), whenever both sides are defined.<\/p>\n

(c) Matrix multiplication is distributive over matrix addition i.e<\/p>\n

(i) A (B + C) = AB + AC<\/p>\n

(ii) (A + B) C = AC + BC whenever both sides of equality are defined.<\/p>\n

(d) If A is an \\(m\\times n\\) matrix, then \\(I_m\\) A = A = A \\(I_n\\).<\/p>\n

(e) If A is \\(m\\times n\\) matrix and O is a null matrix, then<\/p>\n

(i) \\(A_{m\\times n}\\) \\(O_{n\\times p}\\) <\/p>\n

(ii) \\(O_{p\\times m}\\) \\(A_{m\\times n}\\)<\/p>\n

i.e. the product of the matrix with a null matrix is always a null matrix.<\/p>\n

Positive Integral Powers of a Square Matrix<\/h2>\n

for any square matrix, we define<\/p>\n

(i) \\(A^1\\) = A<\/p>\n

(ii) \\(A^{n+1}\\) = \\(A^n\\).A, where n \\(\\in\\) N<\/p>\n

It is evident from this definition that \\(A^2\\) = AA, \\(A^3\\) = \\(A^2\\)A = (AA) A. etc.<\/p>\n

It can be easily shown that<\/p>\n

(i) \\(A^{m}\\)\\(A^{n}\\) = \\(A^{m+n}\\) and,<\/p>\n

(ii) \\((A^{m})^n\\) = \\(A^{mn}\\) for all m, n \\(\\in\\) N.<\/p>\n

Matrix Polynomial<\/h2>\n

Let f(x) = \\(a_0x^n\\) + \\(a_1x^{n-1}\\) + \\(a_2x^{n-2}\\) + ….. + \\(a_{n-1}x\\) + \\(a_n\\) be a polynomial and let A be a square matrix of order n. Then,<\/p>\n

f(A) = \\(a_0A^n\\) + \\(a_1A^{n-1}\\) + \\(a_2A^{n-2}\\) + ….. + \\(a_{n-1}A\\) + \\(a_nI_n\\) <\/p>\n

is called a matrix polynomial.<\/p>\n

for example, if f(x) = \\(x^2\\) – 3x + 2 is a polynomial and A is a square matrix, then f(A) = \\(A^2\\) – 3A + 2I is a matrix polynomial.<\/p>\n\n\n

\n
Next – Transpose of Matrix – Definition & Example<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Multiplication of Matrices – Examples & Definition<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn properties of multiplication of matrices, positive integral powers of square matrix and matrix polynomial. Let’s begin – Also Read : Multiplication of Matrices \u2013 Examples & Definition Properties of Multiplication of Matrices (a) Matrix multiplication is not commutative in general i.e AB \\(\\ne\\) BA. (b) Matrix multiplication is associative i.e. (AB) …<\/p>\n

Properties of Multiplication of Matrices<\/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":[486,487,485],"yoast_head":"\nProperties of Multiplication of Matrices - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn properties of multiplication of matrices and positive integral powers of square matrix and matrix polynomial.\" \/>\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\/properties-of-multiplication-of-matrices\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Properties of Multiplication of Matrices - 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