Problem

The matrix $M$ is given by

$$M = \begin{pmatrix} 2 & 0 & -1 \\ k & 3 & 2 \\ -2 & 1 & k \end{pmatrix}$$

(a) Show that $\det M = 5k - 10$.

Given that $k \ne 2$,

(b) find $M^{-1}$ in terms of $k$.

The points $O(0,0,0)$, $A(4,-8,3)$, $B(-2,5,-4)$ and $C(4,-6,8)$ are the vertices of a tetrahedron $T$. The transformation represented by matrix $M$ transforms $T$ to a tetrahedron with volume $50$.

(c) Determine the possible values of $k$.