From chapter 5 of Steven W. Smith's The Scientist and Engineer's Guide to Digital Signal Processing,
I understand that linearity is commutative according to Smith. If two matrices, $A$ and $B$, each representing a linear system, why is two matrix multiplication not commutative? These two facts seem contradictory. What am I misunderstanding?
This only holds in general for SISO systems. For MIMO systems (akin to matrices), the commutative property does not need to hold. Consider two systems with shapes $1\times 2$ and $2\times 1$, the size of the product is either $1\times 1$ or $2\times 2$ depending on the order.