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If the matrix $\mathrm{B}$ is the adjoint of the square matrix A and $\alpha$ is the value of the determinant of $\mathrm{A}$, then what is $\mathrm{AB}$ equal to ?
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The correct answer is:
$\alpha$
Since, adjoint of the square matrix A is $\mathrm{B} \Rightarrow \frac{\mathrm{B}}{|\mathrm{A}|}=\mathrm{A}^{-1}$
$\Rightarrow \frac{\mathrm{AB}}{|\mathrm{A}|}=\mathrm{AA}^{-1}=\mathrm{I}$
$\Rightarrow \mathrm{AB}=|\mathrm{A}| \mathrm{I} \Rightarrow \mathrm{AB}=\alpha \mathrm{I}$
$\Rightarrow \frac{\mathrm{AB}}{|\mathrm{A}|}=\mathrm{AA}^{-1}=\mathrm{I}$
$\Rightarrow \mathrm{AB}=|\mathrm{A}| \mathrm{I} \Rightarrow \mathrm{AB}=\alpha \mathrm{I}$
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