Question

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

Answer 1

Answer: 729

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