Question

Let $tr\left(X\right)$ and $adj\left(X\right)$ denote the trace and adjoint of a square matrix $X.$ If $M$ is a non-singular square matrix of order $3$ such that $M^{-1}=\left[\begin{array}{ccc}3& 4& 5\\ 4& 5& 3\\ 5& 3& 4\end{array}\right],$ then the value of $|15tr\left(adj\right(M\left)\right)|$ is

Answer 1

$M^{-1}=\left[\begin{array}{ccc}3& 4& 5\\ 4& 5& 3\\ 5& 3& 4\end{array}\right]$
$\Rightarrow |M^{-1}|=3(20-9)-4(16-15)+5(12-25)$
$\Rightarrow |M^{-1}|=-36$
and $tr\left(M^{-1}\right)=3+5+4=12$

We know that $M^{-1}=\frac{adj\left(M\right)}{|M|}$
$\Rightarrow adj\left(M\right)=|M|\cdot M^{-1}$
$\Rightarrow tr(adj\left(M\right))=|M|tr\left(M^{-1}\right)$
$\Rightarrow tr(adj\left(M\right))=\frac{-1}{36}\times 12=-\frac{1}{3}$
$|15tr(adj\left(M\right))|=|15\times (-\frac{1}{3})|=5$