In general, matrix multiplication is notcommutative. Furthermore, in general there is no matrix inverse even when . Finally, can be zero even without or . And when , we may still have , a simple example of which is provided by

Gantmacher, F. R. Ch. 8 in The Theory of Matrices, Vol. 1. Providence, RI: Amer. Math. Soc., 1998.Taussky,
O. "Commutativity in Finite Matrices." Amer. Math. Monthly64,
229-235, 1957.