Order matters in a product

AB may be defined while BA is not. A 2 × 2 matrix times a 2 × 3 matrix is fine, but the 2 × 3 matrix times the 2 × 2 one is not, since 3 ≠ 2.

Both are defined exactly when A is m × n and B is n × m. Then AB is m × m and BA is n × n, so they can even have different orders: for a 2 × 3 A and 3 × 2 B, AB is 2 × 2 and BA is 3 × 3.

Even square matrices of the same order usually fail to commute. With A = [[1, 0], [0, −1]] and B = [[0, 1], [1, 0]], AB = [[0, 1], [−1, 0]] but BA = [[0, −1], [1, 0]].

Some pairs do commute: any two diagonal matrices of the same order, a matrix with I or O, and a matrix with its own powers. [[1, 0], [0, 2]] and [[3, 0], [0, 4]] give [[3, 0], [0, 8]] in either order.

A product of two non-zero matrices can be the zero matrix: [[0, −1], [0, 2]] times [[3, 5], [0, 0]] is O. So AB = O does not force A = O or B = O.

For the same reason AB = AC does not allow cancelling A to get B = C.