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If $\mathrm{D}$ is determinant of order 3 and $\mathrm{D}^{\prime}$ is the determinant obtained by replacing the elements of $D$ by their cofactors, then which one of the following is correct?
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The correct answer is:
$\mathrm{D}^{\prime}=\mathrm{D}^{2}$
$\mathrm{D}^{\prime}=$ cofactor $\mathrm{D}$ $\Rightarrow\left|\mathrm{D}^{\prime}\right|=\mid$ cofactor $\mathrm{D} \mid$
$\Rightarrow\left|\mathrm{D}^{\prime}\right|=|\mathrm{D}|^{3-1}$
$\Rightarrow\left|\mathrm{D}^{\prime}\right|=|\mathrm{D}|^{2}$
So $\mathrm{D}^{\prime}=\mathrm{D}^{2}$
$\Rightarrow\left|\mathrm{D}^{\prime}\right|=|\mathrm{D}|^{3-1}$
$\Rightarrow\left|\mathrm{D}^{\prime}\right|=|\mathrm{D}|^{2}$
So $\mathrm{D}^{\prime}=\mathrm{D}^{2}$
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