The matrix \(\mathbf{A}\) is given by
\(\mathbf{A}=\left(\begin{array}{ccc} a & -6 a & 2 a+2 \\ 0 & 1-a & 0 \\ 0 & 2-a & -1 \end{array}\right)\)
where \(a\) is a constant with \(a \neq 0\) and \(a \neq 1\).
(a) Show that the equation \(\mathbf{A}\left(\begin{array}{c}x \\ y \\ z\end{array}\right)=\left(\begin{array}{l}1 \\ 2 \\ 3\end{array}\right)\) has a unique solution and interpret this situation geometrically.

(b) Show that the eigenvalues of \(\mathbf{A}\) are \(a, 1-a\) and -1 .

(c) Find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \(\mathbf{A}^{4}=\mathbf{P D P}^{-1}\).

(d) Use the characteristic equation of \(\mathbf{A}\) to find \(\mathbf{A}^{4}\) in terms of \(\mathbf{A}\) and \(a\).