(a) Find the values of \(a\) for which the system of equations

\(\begin{aligned}\frac{3}{2}x+3y+8z&=1\\ax+3y+4z&=2\\ay-z&=3\end{aligned}\)

does not have a unique solution.

The matrix \(\mathbf{A}\) is given by

\(\mathbf{A}=\begin{pmatrix}\frac{3}{2}&3&8\\0&3&4\\0&0&-1\end{pmatrix}\).

(b) Given that \(\mathbf{B}=\mathbf{A}^{-1}\), use the characteristic equation of \(\mathbf{A}\) to show that \(\mathbf{B}^{2}=p\mathbf{I}+q\mathbf{A}\), where \(p\) and \(q\) are constants to be determined.

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