{"id":4897,"date":"2021-08-31T17:52:51","date_gmt":"2021-08-31T17:52:51","guid":{"rendered":"https:\/\/mathemerize.com\/?p=4897"},"modified":"2022-01-16T17:01:22","modified_gmt":"2022-01-16T11:31:22","slug":"multiplication-of-matrices","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/multiplication-of-matrices\/","title":{"rendered":"Multiplication of Matrices – Examples and Definition"},"content":{"rendered":"

Here you will learn multiplication of matrices with definition and examples.<\/p>\n

Let’s begin –<\/p>\n

Multiplication of Matrices<\/h2>\n

Definition : <\/strong>Two matrices A and B are conformable for the product AB if the number of columns in A is same as the number of row in B.<\/p>\n

Let matrix A is of order \\(m\\times n\\) then m is the number of rows and n is the number of coumns in A<\/p>\n

and matrix B is of order \\(n\\times p\\) then n is the number of rows and p is the number of columns in B.<\/p>\n

Thus, if A = \\([a_{ij}]_{m\\times n}\\) and B =  \\([b_{ij}]_{n\\times p}\\) are two matrices of order \\(m\\times n\\) and \\(n\\times p\\) respectively, then their product AB is conformable and of order \\(m\\times p\\) and is defined as<\/p>\n

\\((AB)_{ij}\\) = ( \\(i^{th}\\) row of A) (( \\(j^{th}\\) column of B)  for all i = 1, 2, ….., m and j = 1, 2, ……. , p.<\/p>\n

\\(\\implies\\)  \\((AB)_{ij}\\) = \\(\\begin{bmatrix} a_{i1} & a_{i2} & …… & a_{in} \\end{bmatrix}\\) \\(\\begin{bmatrix} b_{1j} \\\\ b_{2j} \\\\ . \\\\ . \\\\ . \\\\ b_{nj}  \\end{bmatrix}\\)<\/p>\n

Note<\/strong> : If A and B are two matrices such that AB exists, then BA may or may not exist.<\/p>\n

Example<\/span><\/strong> :<\/h2>\n

If A = \\(\\begin{bmatrix} 2 & 1 & 3 \\\\ 3 &  -2 & 1 \\\\ -1 & 0 &  1  \\end{bmatrix}\\) and B = \\(\\begin{bmatrix} 1 & -2 \\\\  2 & 1 \\\\ 4 &  -3  \\end{bmatrix}\\), then A is a \\(3\\times 3\\) matrix and B is a \\(3\\times 2\\) matrix. Therefore, A and b are conformable for the product AB and it is of order \\(3\\times 2\\) such that<\/p>\n

\\((AB)_{11}\\) = (first row of A)(first column of B)<\/p>\n

\\(\\implies\\) \\((AB)_{11}\\)  = \\(\\begin{bmatrix} 2 & 1 & 3 \\end{bmatrix}\\) \\(\\begin{bmatrix} 1 \\\\ 2 \\\\  4  \\end{bmatrix}\\)<\/p>\n

= 2*1 + 1*2 + 3*4 = 16<\/p>\n

\\((AB)_{12}\\) = (first row of A)(second column of B)<\/p>\n

\\(\\implies\\) \\((AB)_{12}\\)  = \\(\\begin{bmatrix} 2 & 1 & 3 \\end{bmatrix}\\) \\(\\begin{bmatrix} -2 \\\\ 1 \\\\  3  \\end{bmatrix}\\)<\/p>\n

= 2*(-2) + 1*1 + 3*(-3) = -12<\/p>\n

\\((AB)_{21}\\) = (second row of A)(first column of B)<\/p>\n

\\(\\implies\\) \\((AB)_{21}\\)  = \\(\\begin{bmatrix} 3 & -2 & 1 \\end{bmatrix}\\) \\(\\begin{bmatrix} 1 \\\\ 2 \\\\  4  \\end{bmatrix}\\)<\/p>\n

= 3*1 + (-2)*2 + 1*(4) = 3<\/p>\n

Similarly, we obtain <\/p>\n

\\((AB)_{22}\\) = -11 , \\((AB)_{31}\\) = 3 and \\((AB)_{32}\\) = -1<\/p>\n

\\(\\therefore\\) AB = \\(\\begin{bmatrix} 16 & -12 \\\\ 3 & -11 \\\\ 3 & -1 \\end{bmatrix}\\) <\/p>\n

Note : <\/strong>In this case BA does not exist, because the number of columns in B is not same as the number of rows in A.<\/p>\n\n\n

\n
Next – Properties of Multiplication of Matrices<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Subtraction of Matrices with Examples<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn multiplication of matrices with definition and examples. Let’s begin – Multiplication of Matrices Definition : Two matrices A and B are conformable for the product AB if the number of columns in A is same as the number of row in B. Let matrix A is of order \\(m\\times n\\) then m …<\/p>\n

Multiplication of Matrices – Examples and Definition<\/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":[486,485],"yoast_head":"\nMultiplication of Matrices - Examples and Definition<\/title>\n<meta name=\"description\" content=\"In this post you will learn multiplication of matrices with definition and 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\/multiplication-of-matrices\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Multiplication of Matrices - 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