Change of Summation and Matrix Consider two matrices, $A$, a $m$ by $n$ matrix, and $C$, a $n$ by $p$.
Consider the following operation:$A_{j,r}$ represents entry of row $j$ and column $r$ of matrix $A$.
$\sum\limits_{r=1}^n C_{r,k}(\sum\limits_{j=1}^m A_{j,r}w_j)=\sum\limits_{j=1}^m(\sum\limits_{r=1}^n A_{j,r}C_{r,k})w_j$
$\textbf{My Question:}$ How does this switch of the summation and the matrix entries occur? Should I just try this out with 2 by 2? I don't quite understand.
Reference:
Axler, Sheldon J. $\textit{Linear Algebra Done Right}$, New York: Springer, 2015.
 A: If you observe the given summations, you will soon observe that the given equality is actually the associative property of matrix multiplication. 
Let $w$ be a $1×m$ row matrix, with $w_j$ denoting the entry in its $j^{th}$ column. Then, the given equality is simply the statement
$(wA)C = w(AC)$
Where the product is a $1×p$ matrix.
Still, let's see what actually happened that the sigmas switched.
$\sum\limits_{r=1}^n C_{r,k}(\sum\limits_{j=1}^m A_{j,r}w_j)$
$=\sum\limits_{r=1}^n C_{r,k}(A_{1,r}w_1 + A_{2,r}w_2 + ... + A_{m,r}w_m)$
$=C_{1,k}(A_{1,1}w_1 + A_{2,1}w_2 + ... + A_{m,1}w_m)$
$+ C_{2,k}(A_{1,2}w_1 + A_{2,2}w_2 + ... + A_{m,2}w_m) +...$
$+C_{n,k}(A_{1,n}w_1 + A_{2,n}w_2 + ... + A_{m,n}w_m)$
Now, we take out the $w_i$'s common in place of the $C_{r,k}$'s.
$=w_1(A_{1,1}C_{1,k} + A_{1,2}C_{2,k} + ... + A_{1,n}C_{n,k})$
$+w_2(A_{2,1}C_{1,k} + A_{2,2}C_{2,k} + ... + A_{2,n}C_{n,k}) +...$
$+w_m(A_{m,1}C_{1,k} + A_{m,2}C_{2,k} + ... + A_{m,n}C_{n,k})$
$=\sum\limits_{j=1}^m w_j(A_{j,1}C_{1,k} + A_{j,2}C_{2,k} + ... +A_{j,n}C_{n,k})$
$=\sum\limits_{j=1}^mw_j(\sum\limits_{r=1}^n A_{j,r}C_{r,k})$
This is how the sigma switches in any case where there are two nested sigmas. We just change the variable we were factorising out.
Hope it helped
