8 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { r r r } a & 2 & - 1
2 & 3 & - 1
2 & - 1 & 1 \end{array} \right)\), where \(a\) is a constant.
- Show that the determinant of \(\mathbf { M }\) is \(2 a\).
- Given that \(a \neq 0\), find the inverse matrix \(\mathbf { M } ^ { - 1 }\).
- Hence or otherwise solve the simultaneous equations
$$\begin{array} { r }
x + 2 y - z = 1
2 x + 3 y - z = 2
2 x - y + z = 0
\end{array}$$ - Find the value of \(k\) for which the simultaneous equations
$$\begin{array} { r }
2 y - z = k
2 x + 3 y - z = 2
2 x - y + z = 0
\end{array}$$
have solutions. - Do the equations in part (iv), with the value of \(k\) found, have a solution for which \(x = z\) ? Justify your answer.