Question
Let \(A\) and \(B\) be square matrices of order \(3 \times 3\). Is \[ (AB)^2 = A^2 B^2? \] Give reasons.
Solution
Step 1: Expand LHS
\[ (AB)^2 = (AB)(AB) \] \[ = A(BA)B \]Step 2: Compare with RHS
\[ A^2 B^2 = AABB \]Step 3: Key Observation
\[ A(BA)B \ne A(AB)B \] because: \[ AB \ne BA \quad \text{(in general)} \]Step 4: Conclusion
\[ (AB)^2 \ne A^2 B^2 \] unless: \[ AB = BA \]Final Answer
\[
(AB)^2 = A(BA)B \ne A^2 B^2 \quad \text{(in general)}
\]
\[
\text{Because } AB \ne BA
\]
\[
\text{Equality holds only if } AB = BA
\]