8 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 \(\lambda \mu\). State the eigenvalues of the matrix \(\mathbf { C }\), where $$\mathbf { C } = \left( \begin{array} { r r r } - 1 & - 1 & 3
0 & 1 & 2
0 & 0 & 2 \end{array} \right) ,$$ and find corresponding eigenvectors. Show that \(\left( \begin{array} { l } 1
6
3 \end{array} \right)\) is an eigenvector of the matrix \(\mathbf { D }\), where $$\mathbf { D } = \left( \begin{array} { r r r } 1 & - 1 & 1
- 6 & - 3 & 4
- 9 & - 3 & 7 \end{array} \right) ,$$ and state the corresponding eigenvalue. Hence state an eigenvector of the matrix CD and give the corresponding eigenvalue.