$$\sum_{r = 1}^n \sum_{p=1}^n A_{j,p}A_{p,r}A_{r,k}=\sum_{p = 1}^n \sum_{r=1}^n A_{j,p}A_{p,r}A_{r,k}$$
Would reviewing double sums help me to avoid asking questions like this?
Does anybody know of a good source I can use to learn properties of double sums like this?
On
There are only two summation indices, namely $r$ and $p.$ Hence you may think of the terms of the sum as arranged in a rectangular array $(p,r)$ of order $n×n$
On LHS we vary $p$ first and fix $r.$ Then we vary $r.$ This means for each column, we add all its components, and then add all the column-sums. On RHS the reverse is done; we vary $r$ first, fixing $p.$ Then vary $p.$ This means for each fixed row we add all its components, them add all such sums. Of course we get the same result however we choose to add. Thus, $\text{LHS}=\text{RHS}.$
On
A good introduction answering many of this and related questions is provided in chapter 2: Sums of Concrete Mathematics by R.L. Graham, D.E. Knuth and O. Patashnik. Double sums are treated in section 2.4 Multiple Sums.
The short answer is that addition is commutative. More generally, $$\sum_{i=1}^m \sum_{j=1}^n c_{i,j} = \sum_{j=1}^n \sum_{i=1}^m c_{i,j}.$$ Just think of adding up the elements of an $m \times n$ matrix either row-wise or column-wise.