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 \(\mu\). Show that \(\mathbf { e }\) is an eigenvector of the matrix \(\mathbf { A B }\) with corresponding eigenvalue \(\lambda \mu\).
Find the eigenvalues and corresponding eigenvectors of the matrix \(\mathbf { A }\), where
$$\mathbf { A } = \left( \begin{array} { r r r }
0 & 1 & 3
3 & 2 & - 3
1 & 1 & 2
\end{array} \right) .$$
The matrix \(\mathbf { B }\), where
$$\mathbf { B } = \left( \begin{array} { r r r }
3 & 6 & 1
1 & - 2 & - 1
6 & 6 & - 2
\end{array} \right) ,$$
has eigenvectors \(\left( \begin{array} { r } 1 - 1 0 \end{array} \right) , \left( \begin{array} { r } 1 - 1 1 \end{array} \right)\) and \(\left( \begin{array} { l } 1 0 1 \end{array} \right)\). Find the eigenvalues of the matrix \(\mathbf { A B }\), and state corresponding eigenvectors.