The currently standard matrix multiplication is isomorphic to split-quaternions (for 2x2 matrices, and similar for higher ranks, maybe this is called Clifford algebras). As such, the sets of square matrices is not commutative, has zero divisors and nilpotents.
I wonder, why other matrix multiplication rules were not chosen. For instance, the tessarine-like that would be commutative?
Yes, usual quaternions or tessarines or split-numbers of any dimension can be represented by the matrices, but since the multiplication rules differ, one would need martices of 2x rank to represent them.