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) (C
D) -> C on top of D
= AC + BD

Answer 1

I think there is a mistake in the problem, for example, multiplying the m x r matrix $A$ by the m x s matrix $B$.

Let $A$ be 2x3 matrix and $B$ be 2x1 matrix, then the product of $A$ by $B$ won't be defined:

$\begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{pmatrix}$ *$\begin{pmatrix} b_{11} \\ b_{21} \end{pmatrix}$ $=?$

To multiply matrices, the number of columns of one of them must be equal to the number of rows of another one!

For example,

$\begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ a_{31} & a_{32} \end{pmatrix}$ * $\begin{pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{pmatrix}$ =$\begin{pmatrix} a_{11}*b_{11}+ a_{12}*b_{21}& a_{11}*b_{12}+a_{12}*b_{22} \\ a_{21}*b_{11}+ a_{22}*b_{21}& a_{21}*b_{12}+a_{22}*b_{22} \\ a_{31}*b_{11}+ a_{32}*b_{21}& a_{31}*b_{12}+a_{32}*b_{22} \end{pmatrix}$

In matrices $A$ and $B$ we can't do like this!