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.