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If $A$ and $B$ are $2$ different matrices satisfying $A^{3} = B^{3}$ and $A^{2} B = B^{2} A,$ then value of det $(A^{2} + B^{2})$ is

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Solution

Given $A^{3} = B^{3}$ and $A^{2} B = B^{2} A,$ ... $(1)$
Consider $(A^{2} + B^{2}) (A - B) = A^{3} - A^{2} B + B^{2} A - B^{3}$
$\Rightarrow (A^{2} + B^{2}) (A - B) = O$ .... $(2)$
Since, $A$ and $B$ are different matrices
So, $A \neq B$
$A - B \neq O$
So, from equation $(2)$ , it follows
$A^{2} + B^{2} = O$
$\Rightarrow d e t (A^{2} + B^{2}) = 0$

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