Exercise 2.4.37

The proof of $(\mathit{AB})c=A(\mathit{Bc})$ used the column rule for matrix multiplication. "The same is true for all other columns of $C$."

Even for nonlinear transformations, $A(B(c))$ would be the "composition" of $A$ with $B$ (applying $B$ then $A$). This composition $A\circ B$ is just written as $\mathit{AB}$ for matrices.

One of many uses for the associative law: The left-inverse $B=$ the right-inverse $C$ because of $B=B(\mathit{AC})=(\mathit{BA})C=C$.