# Let A and B be square matrices of the order 3 × 3.

Question:

Let $A$ and $B$ be square matrices of the order $3 \times 3$. Is $(A B)^{2}=A^{2} B^{2} ?$ Give reasons.

Solution:

Yes, $(A B)^{2}=A^{2} B^{2}$ if $A B=B A$

If $A B=B A$, then

$(A B)^{2}=(A B)(A B)$

$=A(B A) B$            (associative law)

$=A(A B) B$

$=A^{2} B^{2}$