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If the determinant of the adjoint of a (real) matrix of order 3 is 25 , then the determinant of the inverse of the matrix is
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$\pm 0.2$
Given, the determinant of the adjoint of a (real) matrix of order 3 is $25 .$
i.e., $\quad|\operatorname{adj} A|=25...(i)$
We know that,
$|\operatorname{adj} A|=|A|^{n-1} \quad$ (here, $n=3$ )
$\Rightarrow \quad|A|^{3-1}=|A|^{2}=25 \quad$ [from Eq. (i)]
$\Rightarrow \quad|A|=\pm 5$
$\therefore \quad\left|A^{-1}\right|=|A|^{-1}=\frac{1}{|A|}=\pm \frac{1}{5}=\pm 0.2$ (by property)
i.e., $\quad|\operatorname{adj} A|=25...(i)$
We know that,
$|\operatorname{adj} A|=|A|^{n-1} \quad$ (here, $n=3$ )
$\Rightarrow \quad|A|^{3-1}=|A|^{2}=25 \quad$ [from Eq. (i)]
$\Rightarrow \quad|A|=\pm 5$
$\therefore \quad\left|A^{-1}\right|=|A|^{-1}=\frac{1}{|A|}=\pm \frac{1}{5}=\pm 0.2$ (by property)
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