Question

If $A$ be a square matrix of order $3$ , such that $\left|A\right|=\sqrt{5}$ , and $\left|A d j \left(- 3 A^{- 2}\right)\right|=\lambda $ , then $25\lambda =$

Answer 1

$\left|A d j \left(- 3 A^{- 2}\right)\right|=\left(\left|\left(- 3 A^{- 2}\right)\right|\right)^{2}=3^{6}\left(\left|A^{- 2}\right|\right)^{2}=3^{6}\times \frac{1}{\left(\left|A\right|\right)^{4}}$
$\lambda =3^{6}\times \frac{1}{25}=\frac{729}{25}$
$\Rightarrow 25\lambda =729$