Let P = [3, -1, -2; 2, 0, α; 3, -5, 0], where α ∈ R. Suppose Q =[ q_i j] is a matrix satisfying PQ = kI_3 for some non-zero k ∈ R. If q_23=-k/8 and |…

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