Question

Let $A$ be a non-singular matrix of order $3$ such that $Aadj\left(3 A\right)=5AA^{T}$ , then $5\left(\sqrt[3]{\left|A^{- 1}\right|}\right)$ is equal to

Answer 1

$A(\operatorname{adj}(3 A))=5 A A^{T}$
Taking determinant on both sides, we get,
$\left|A\right|\left|a d j 3 A\right|=\left|5 A A^{T}\right|$
$\Rightarrow \left|A\right|\left|3 A\right|^{2}=5^{3}\left|A\right|^{2}$
$\Rightarrow 3^{6}\left(\left|A\right|\right)^{3}=5^{3}\left(\left|A\right|\right)^{2}\Rightarrow \left|A\right|=0,\frac{125}{729}\left(\left|A\right| \neq 0\right)$
Hence, $\sqrt[3]{\left|A^{- 1}\right|}=\left(\frac{1}{\left|A\right|}\right)^{1/3}=\frac{9}{5}$
$\Rightarrow 5\left(\sqrt[3]{\left|A^{- 1}\right|}\right)=9$