Is it always true that AB = BA?

NO! Even when both products can be formed they are rarely equal.

For example, calculate both of the following products (you might wish to use paper and pencil):

The results show that in general for matrices A and B, AB =/= BA.

Is A_{4×4} 0_{4×4}== 0_{4×4} A_{4×4}, where 0 is the zero matrtix?