Not sure why this is true about matrices, but this isn't if they are commutative Let $A$, $B$ and $C$ be three matrices.
Although for general matrices $PQ \ne QP$, in this particular case I am told that $AC=CA$.
I am also told that $A(B+C) \ne BA + CA$.
If I am being told in part of the question that the matrices are commutative when being multiplied (as in $AC = CA$), why isn't $A(B+C) = BA + CA$?

You are given that $AC = CA$ i.e. that $A$ and $C$ commmute. You are not told that $A$ commutes with every matrix. Therefore, we cannot conclude that $A(B+C) = BA + CA$. In fact, the claim that $A(B+C) \ne BA + CA$ is perfectly consistent with the given data. Indeed, distributivity gives $$A(B+C) = AB + AC$$
and then commutativity of $A$ and $C$ gives $$A(B+C) = AB + CA.$$
Since matrix addition is invertible, the inequality $AB + CA \ne BA + CA$ is equivalent to $AB \ne BA$ (just subtract $CA$ from both sides). Thus, $A$ and $B$ do not commute. This is fine. It is very easy to find three matrices that act like this: take two matrices that don't commute, and let $C = I$ be the identity matrix.
