Let $P=\left[\begin{array}{ccc}3& -1& -2\\ 2& 0& \alpha \\ 3& -5& 0\end{array}\right],$where$\alpha \in R.$  Suppose $Q=\left[q_{if}\right]$ is a matrix satisfying $PQ=k{I}_3$  for some non-zero If  $\left|Q\right|=\frac{k^2}{2},$ and$q_{23}=-\frac{k}{8}$ then ${\alpha }^2+k^2$ is equal to 

$Q=\frac{k}{\left|P\right|}\left(adjP\right)I\Rightarrow Q=\frac{k}{\left(20+12\alpha \right)}\left[\begin{array}{ccc}-& -& -\\ -& -& \left(-3\alpha -4\right)\\ -& -& -\end{array}\right]\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

$\begin{array}{c}\because q_{23}=-\frac{k}{8}\\ =\frac{k}{\left(20+12\alpha \right)}\left(-3\alpha -4\right)\\ 2\left(3\alpha +4\right)=-\frac{k}{8}\\ 5+3\alpha =-\frac{k}{8}\end{array}$

$3\alpha =-3\Rightarrow \alpha =-1$

$\left|Q\right|=\frac{k^3\left|I\right|}{\left|P\right|}\Rightarrow \frac{k^2}{2}=\frac{k^3}{\left(20+12\alpha \right)}$

$\left(20+12\alpha \right)=2k\Rightarrow 8=2k\Rightarrow k=4$