8 The matrix \(\mathbf { A }\) is given by
$$\mathbf { A } = \left( \begin{array} { c c c }
a & - 6 a & 2 a + 2
0 & 1 - a & 0
0 & 2 - a & - 1
\end{array} \right)$$
where \(a\) is a constant with \(a \neq 0\) and \(a \neq 1\).
- Show that the equation \(\mathbf { A } \left( \begin{array} { c } x
y
z \end{array} \right) = \left( \begin{array} { c } 1
2
3 \end{array} \right)\) has a unique solution and interpret this situation geometrically. - Show that the eigenvalues of \(\mathbf { A }\) are \(a , 1 - a\) and - 1 .
- Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { 4 } = \mathbf { P D P } ^ { - 1 }\).
- Use the characteristic equation of \(\mathbf { A }\) to find \(\mathbf { A } ^ { 4 }\) in terms of \(\mathbf { A }\) and \(a\).
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