Multiplication of a Matrix

Multiplication of a matrix

Let A and B matrices of orders m x n and n x p respectively. The product AB of

matrices A and B is defined only if the number of columns of A is equal to the number

of rows of B . The product AB will be a m x p matrix.

In other words, if $A=[a_{ij}]_{m\times n}$ and $B=[b_{kj}]_{n\times p}$ then the matrix

$AB=[c_{ij}]_{m\times p} \ ,\ where\ c_{ij}=a_{i1}b_{1j}+a_{i2}b_{2j}+...+a_{in}b_{nj}$    $\ , \ 1\leq i\leq m \ , \ 1\leq j\leq p$
In the product AB , A  is called pre-factor of AB and B , the post-factor of AB .

For example :
$let \ A = \begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22} \end{bmatrix}\ and \ B = \begin{bmatrix} b_{11} & b_{12} & b_{13}\\ b_{21} & b_{22} & b_{23} \end{bmatrix}$
Here the product AB is defined, because the number of columns (=2) of A equals the
number of rows (=2) 0f B . In this case,

$AB=\begin{vmatrix} a_{11} &a_{12} \\ a_{21} & a_{22} \end{vmatrix}\begin{vmatrix} b_{11} & b_{12} &b_{13} \\ b_{21} & b_{22} & b_{23} \end{vmatrix}$

$=\begin{vmatrix} a_{11}b_{11}+a_{12}b_{21} & a_{11}b_{12}+a_{12}b_{22} & a_{11}b_{13}+a_{12}b_{23} \\ a_{21}b_{11}+a_{22}b_{21} & a_{21}b_{12}+a_{22}b_{22} & a_{21}b_{13}+a_{22}b_{23} \end{vmatrix}$

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Multiplication of a Matrix

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