(a) It is given that \(\lambda\) is an eigenvalue of the non-singular square matrix \(\mathbf{A}\), with corresponding eigenvector \(\mathbf{e}\).

Show that \(\mathbf{e}\) is an eigenvector of \(\mathbf{A}^{3}\) with corresponding eigenvalue \(\lambda^{3}\).

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

\(\mathbf{A}=\begin{pmatrix}-1&3&4\\0&1&0\\0&-2&5\end{pmatrix}\).

(b) Show that the eigenvalues of \(\mathbf{A}\) are \(-1\), \(1\) and \(5\).

(c) Find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \(\mathbf{A}-2\mathbf{I}=\mathbf{P}\mathbf{D}\mathbf{P}^{-1}\).

(d) Use the characteristic equation of \(\mathbf{A}\) to show that \((\mathbf{A}-2\mathbf{I})^{3}=a\mathbf{A}^{2}+b\mathbf{A}+c\mathbf{I}\), where \(a\), \(b\) and \(c\) are constants to be determined.