Let A and B be square matrices of the order 3 × 3. Is (AB)2 = A2B2 ? Give reasons.
We are given that,
A and B are square matrices of the order 3 × 3.
We need to check whether (AB)2 = A2B2 is true or not.
Take (AB)2.
(AB)2 = (AB)(AB)
[∵ A and B are of order (3 × 3) each, A and B can be multiplied; A and B be any matrices of order (3 × 3)]
⇒ (AB)2 = ABAB
[∵ (AB)(AB) = ABAB]
⇒ (AB)2 = AABB
[∵ ABAB = AABB; as A can be multiplied with itself and B can be multiplied by itself]
⇒ (AB)2 = A2B2
So, note that, (AB)2 = A2B2 is possible.
But this is possible if and only if BA = AB.
And BA = AB is always true whenever A and B are square matrices of any order. And for BA = AB,
(AB)2 = A2B2