(a) Find the values of \(a\) for which the system of equations
\[\begin{array}{r}
3 x+y+z=0 \\
a x+6 y-z=0 \\
a y-2 z=0
\end{array}\]
does not have a unique solution.

The matrix \(\mathbf{A}\) is given by
\[\mathbf{A}=\left(\begin{array}{rrr}
3 & 1 & 1 \\
0 & 6 & -1 \\
0 & 0 & -2
\end{array}\right)\]
(b) Use the characteristic equation of \(\mathbf{A}\) to find the inverse of \(\mathbf{A}^{2}\).
(c) Find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \(\mathbf{A}^{5}=\mathbf{P D P}^{-1}\).