EITHER

The vector \(\mathbf e\) is an eigenvector of the matrix \(\mathbf A\), with corresponding eigenvalue \(\lambda\), and is also an eigenvector of the matrix \(\mathbf B\), with corresponding eigenvalue \(\mu\).

(i) Show that \(\mathbf e\) is an eigenvector of the matrix \(\mathbf{AB}\) with corresponding eigenvalue \(\lambda\mu\).

(ii) Find the eigenvalues and corresponding eigenvectors of

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

(iii) The matrix

\(\mathbf B=\begin{pmatrix}3&6&1\\1&-2&-1\\6&6&-2\end{pmatrix}\)

has eigenvectors \(\begin{pmatrix}1\\-1\\0\end{pmatrix}\), \(\begin{pmatrix}1\\-1\\1\end{pmatrix}\) and \(\begin{pmatrix}1\\0\\1\end{pmatrix}\). Find the eigenvalues of \(\mathbf{AB}\), and state the corresponding eigenvectors.