Let A and B be square matrices of the order 3 × 3. Is (AB) 2 = A 2 B 2 ? Give reasons.

Let A and B be square matrices of the order 3 × 3. Is (AB)2 = A2B2? Give reasons.

Given that A and B are square matrices of the order 3 × 3.

We know (AB)2 = (AB)(AB)

⇒ (AB)2 = A × B × A × B

⇒ (AB)2 = A(BA)B

If the matrices A and B satisfy the commutative property for multiplication, then AB = BA.

We found (AB)2 = A(BA)B.

Hence, when AB = BA, we have (AB)2 = A(AB)B.

⇒ (AB)2 = A × A × B × B

⇒ (AB)2 = A2B2

Therefore, (AB)2 = A2B2 holds only when AB = BA.

Thus, (AB)2 ≠ A2B2 unless the matrices A and B satisfy the commutative property for multiplication.