8 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { l l l } a & 4 & 2
1 & a & 0
1 & 2 & 1 \end{array} \right)\).
- Find, in terms of \(a\), the determinant of \(\mathbf { M }\).
- Hence find the values of \(a\) for which \(\mathbf { M }\) is singular.
- State, giving a brief reason in each case, whether the simultaneous equations
$$\begin{aligned}
a x + 4 y + 2 z & = 3 a
x + a y & = 1
x + 2 y + z & = 3
\end{aligned}$$
have any solutions when
(a) \(a = 3\),
(b) \(a = 2\).