Q40 / 75 · NTA Abhyas (JEE Main) 2020 · Mathematics · Matrices

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

A
$3$
B
$4$
C
$5$
Correct answer
D
$7$

Solution

{(A B)}^{3} = (A B) (A B) (A B)

= A (B A) (B A) B

= A (A B^{2}) (A B^{2}) B = A^{2} B (B A) B^{3}

= A^{2} B (A B^{2}) B^{3}

= A^{2} (B A) B^{5} = A^{2} (A B^{2}) B^{5}

= A^{3} B^{7}

Hence, the correct answer is option: 'D'.

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