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  4. Let A and B are square matrices of order 3 such that AB2=BA and BA2=AB. If (A B)3=A3Bm , then m is equal to

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Q. Let $A$ and $B$ are square matrices of order $3$ such that $AB^{2}=BA$ and $BA^{2}=AB.$ If $\left(A B\right)^{3}=A^{3}B^{m}$ , then $m$ is equal to

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Solution:

$\left(A B\right)^{3}=\left(A B\right)\left(A B\right)\left(A B\right)$
$=A\left(B A\right)\left(B A\right)B$
$=A\left(A B^{2}\right)\left(A B^{2}\right)B=A^{2}B\left(B A\right)B^{3}$
$=A^{2}B\left(A B^{2}\right)B^{3}$
$=A^{2}\left(B A\right)B^{5}=A^{2}\left(A B^{2}\right)B^{5}$
$=A^{3}B^{7}$