What is the reasoning behind changing the order of summation in this example?

Question

I stumbled upon this while reading a matrix multiplication associativity proof $A(BC)=(AB)C$.
I don't understand how you can change the order of summation thus can't understand the proof.
I attached an image and the part I don't understand is the seccond row. I would be thankful if someone could explain why and how changing the order of summation like this can be done.

Answer 1

These are finite sums, so the order of summation is always fine to change so long as we are summing over the exact same indices ($a+b = b+a$ extends inductively to any rearrangement of a finite sum). In this case the indices are “all values of $k$ between $1$ and $p$, paired with all values of $j$ between $1$ and $n$”. It makes no difference what order you take those two conditions in, especially as neither one has a dependent clause that depends on the specific value of the other (like if $j$ ranged up to $k$ instead of $n$).