(a) Find the value of \(a\) for which the system of equations
\(\begin{array}{c} 3 x+a y=0 \\ 5 x-y=0 \\ x+3 y+2 z=0 \end{array}\)
does not have a unique solution.

The matrix \(\mathbf{A}\) is given by
\(\mathbf{A}=\left(\begin{array}{rrr} 3 & 0 & 0 \\ 5 & -1 & 0 \\ 1 & 3 & 2 \end{array}\right)\)
(b) Find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \(\mathbf{A}^{2}=\mathbf{P D P}^{-1}\).

(c) Use the characteristic equation of \(\mathbf{A}\) to show that
\((\mathbf{A}+6 \mathbf{I})^{2}=\mathbf{A}^{4}(\mathbf{A}+b \mathbf{I})^{2}\)
where \(b\) is an integer to be determined.