Question

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

Answer 1

Answer: 9

$A(\mathrm{adj}(3A))=5A{A}^T$
Taking determinant on both sides, we get,
$\left|A\right|\left|adj3A\right|=\left|5A{A}^T\right|$
$\Rightarrow \left|A\right|{\left|3A\right|}^2=5^3{\left|A\right|}^2$
$\Rightarrow 3^6{\left(\left|A\right|\right)}^3=5^3{\left(\left|A\right|\right)}^2\Rightarrow \left|A\right|=0,\frac{125}{729}\left(\left|A\right|\ne 0\right)$
Hence, $\sqrt[3]{\left|A^{-1}\right|}={\left(\frac{1}{|A|}\right)}^{1/3}=\frac{9}{5}$
$\Rightarrow 5\left(\sqrt[3]{\left|A^{-1}\right|}\right)=9$