Question

Let A be an m x r matrix, B be an m x s matrix, C be an r x n matrix, and D be an s x n matrix.

Prove that

(A B) where A and B are side by side in the 1 x 2 bracket, multiply (C D) where C is on top of D in the 2x1 bracket = AC + BD

Answer 1

We are given that there is one matrix of order $1\times2$ $\ \ \begin{pmatrix} a & b \end{pmatrix}$ and other matrix of order 2$\times$1 $\begin{pmatrix} c \\ d \end{pmatrix}$

We have to prove that their product sum is equal to ($ac+bd$ )

To multiply matrices, the number of columns of one of them must be equal to the number of rows of another one , here this condition is fulfilled and their product will be a 1x 1 matrix

let's multiply

$\begin{pmatrix} a & b\end{pmatrix}\cdot \begin{pmatrix} c\\ d \end{pmatrix}$$=a*c+b*d$

We obtained this result.