$$\sum_{r = 1}^n \sum_{p=1}^n A_{j,p}A_{p,r}A_{r,k}=\sum_{p = 1}^n \sum_{r=1}^n A_{j,p}A_{p,r}A_{r,k}$$
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?
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.
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.
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, $\text{LHS}=\text{RHS}.$
