question_answer

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

Answer:

\[LHS={{(AB)}^{2}}=(AB)\,(AB)\]       [where, \[AB=BA\]] then we can write \[{{(AB)}^{2}}=A\cdot AB\cdot B={{A}^{2}}{{B}^{2}}\] \[\therefore \]      \[{{(AB)}^{2}}={{A}^{2}}{{B}^{2}},\]when \[AB=BA\]