If A, B are, respectively m × n, k × l matrices, then both AB and BA are defined
if and only if n = k and l = m In particular, if both A and B are square matrices of the
same order, then both AB and BA are defined Non-commutativity of multiplication of matrices
Now, we shall see by an example that even if AB and BA are both defined, it is not
necessary that AB = BA Example 13 If 
2 3
1
2
3
A
and B
4 5
4
2
5
2 1


−




=
=




−





, then find AB, BA