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If $A$ and $B$ are square matrices of the same order such that $(A+B)(A-B)=A^{2}-B^{2}$, then $\left(A B A^{-1}\right)^{2}$ is equal to
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The correct answer is:
$\mathrm{B}^{2}$
Given, $(A+B)(A-B)=A^{2}-B^{2}$ $\Rightarrow \quad A^{2}-A B+B A-B^{2}=A^{2}-B^{2}$
$\Rightarrow \quad \mathrm{AB}=\mathrm{BA}$
Now, $\quad\left(\mathrm{ABA}^{-1}\right)^{2}=\left(\mathrm{BAA}^{-1}\right)^{2}=\mathrm{B}^{2}$
$\Rightarrow \quad \mathrm{AB}=\mathrm{BA}$
Now, $\quad\left(\mathrm{ABA}^{-1}\right)^{2}=\left(\mathrm{BAA}^{-1}\right)^{2}=\mathrm{B}^{2}$
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