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.

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.

\hfill \mbox{\textit{CAIE FP1 2011 Q8}}