Find the values of \(a\) for which the system of equations
$$\begin{aligned}
3 x + y + z & = 0
a x + 6 y - z & = 0
a y - 2 z & = 0
\end{aligned}$$
does not have a unique solution.
The matrix \(\mathbf { A }\) is given by
$$\mathbf { A } = \left( \begin{array} { r r r }
3 & 1 & 1
0 & 6 & - 1
0 & 0 & - 2
\end{array} \right) .$$
Use the characteristic equation of \(\mathbf { A }\) to find the inverse of \(\mathbf { A } ^ { 2 }\).
Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { 5 } = \mathbf { P D P } ^ { - 1 }\).
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