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  • Let A and B be square matrices of the order 3 × 3. Is (AB)^2 = A^2B^2? Give reasons.

Let A and B be square matrices of the order 3 × 3. Is (AB)^2 = A^2B^2? Give reasons.

Answers (1)

We have been given that, 

A and B are square matrices of the order 3 \times 3.

We need to check whether (AB)\textsuperscript{2} = A\textsuperscript{2}B\textsuperscript{2}  is true or not.

Take (AB)\textsuperscript{2}.

(AB)\textsuperscript{2} = (AB)(AB)

[\because  A and B are of order (3 \times 3) each, A and B can be multiplied; A and B be any matrices of order (3 \times 3)]

\Rightarrow (AB)\textsuperscript{2} = ABAB

[\because  (AB)(AB) = ABAB]

\Rightarrow (AB)\textsuperscript{2} = AABB  [ if BA = AB]

\Rightarrow (AB)\textsuperscript{2} = A\textsuperscript{2}B\textsuperscript{2}

(AB)\textsuperscript{2} = A\textsuperscript{2}B\textsuperscript{2} is possible if BA = AB.

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