Let $P = \left[\begin{array}{ccc} 3 & -1 & -2 \\ 2 & 0 & \alpha \\ 3 & -5 & 0 \end{array}\right]$, where $\alpha \in \mathbb{R}$. Suppose $Q = [q_{ij}]$ is a matrix such that $PQ = kI$, where $k \in \mathbb{R}, k \neq 0$ and $I$ is the identity matrix of order 3. If $q_{23} = -\frac{k}{8}$ and $\det(Q) = \frac{k^2}{2}$, then\\
(A) $\alpha = 0, k = 8$\\
(B) $4\alpha - k + 8 = 0$\\
(C) $\det(P\operatorname{adj}(Q)) = 2^9$\\
(D) $\det(Q\operatorname{adj}(P)) = 2^{13}$