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JEE Advanced 2016 Paper 1 Mathematics Question 44

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 \(q_{23} = - \displaystyle\frac{k}{8}\) and \(\text{det}(Q) = \displaystyle\frac{k^2}{2}\), then 

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