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