4. The matrix \(\mathbf { M }\) is given by
$$\left( \begin{array} { r r r }
2 & 0 & - 1
k & 3 & 2
- 2 & 1 & k
\end{array} \right)$$
- Show that \(\operatorname { det } \mathbf { M } = 5 k - 10\)
Given that \(k \neq 2\)
- find \(\mathbf { M } ^ { - 1 }\) 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 \(\mathbf { M }\) transforms \(T\) to a tetrahedron with volume 50
- Determine the possible values of \(k\).