6 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { l l l } 4 & - 5 & 3 \\ 3 & - 4 & 3 \\ 1 & - 1 & 2 \end{array} \right)$$ Show that \(\mathbf { e } = \left( \begin{array} { l } 1 \\ 1 \\ 1 \end{array} \right)\) is an eigenvector of \(\mathbf { A }\) and state the corresponding eigenvalue. Find the other two eigenvalues of \(\mathbf { A }\). The matrix \(\mathbf { B }\) is given by $$\mathbf { B } = \left( \begin{array} { r r r } - 1 & 4 & 0 \\ - 1 & 3 & 1 \\ 1 & - 1 & 3 \end{array} \right)$$ Show that \(\mathbf { e }\) is an eigenvector of \(\mathbf { B }\) and deduce an eigenvector of the matrix \(\mathbf { A B }\), stating the corresponding eigenvalue.

6 The matrix $\mathbf { A }$ is given by

$$\mathbf { A } = \left( \begin{array} { l l l } 
4 & - 5 & 3 \\
3 & - 4 & 3 \\
1 & - 1 & 2
\end{array} \right)$$

Show that $\mathbf { e } = \left( \begin{array} { l } 1 \\ 1 \\ 1 \end{array} \right)$ is an eigenvector of $\mathbf { A }$ and state the corresponding eigenvalue.

Find the other two eigenvalues of $\mathbf { A }$.

The matrix $\mathbf { B }$ is given by

$$\mathbf { B } = \left( \begin{array} { r r r } 
- 1 & 4 & 0 \\
- 1 & 3 & 1 \\
1 & - 1 & 3
\end{array} \right)$$

Show that $\mathbf { e }$ is an eigenvector of $\mathbf { B }$ and deduce an eigenvector of the matrix $\mathbf { A B }$, stating the corresponding eigenvalue.

\hfill \mbox{\textit{CAIE FP1 2013 Q6}}