10 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } 2 & 2 & - 3 \\ 2 & 2 & 3 \\ - 3 & 3 & 3 \end{array} \right)$$ The matrix \(\mathbf { A }\) has an eigenvector \(\left( \begin{array} { r } 1 \\ - 1 \\ 1 \end{array} \right)\). Find the corresponding eigenvalue. The matrix \(\mathbf { A }\) also has eigenvalues 4 and 6. Find corresponding eigenvectors. Hence find a matrix \(\mathbf { P }\) such that \(\mathbf { A } = \mathbf { P D P } \mathbf { P } ^ { - 1 }\), where \(\mathbf { D }\) is a diagonal matrix which is to be determined. The matrix \(\mathbf { B }\) is such that \(\mathbf { B } = \mathbf { Q A Q } ^ { - 1 }\), where $$\mathbf { Q } = \left( \begin{array} { r r r } 4 & 11 & 5 \\ 1 & 4 & 2 \\ 1 & 2 & 1 \end{array} \right)$$ By using the expression \(\mathbf { P D P } ^ { - 1 }\) for \(\mathbf { A }\), find the set of eigenvalues and a corresponding set of eigenvectors for \(\mathbf { B }\).
[0pt] [Question 11 is printed on the next page.]

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

$$\mathbf { A } = \left( \begin{array} { r r r } 
2 & 2 & - 3 \\
2 & 2 & 3 \\
- 3 & 3 & 3
\end{array} \right)$$

The matrix $\mathbf { A }$ has an eigenvector $\left( \begin{array} { r } 1 \\ - 1 \\ 1 \end{array} \right)$. Find the corresponding eigenvalue.

The matrix $\mathbf { A }$ also has eigenvalues 4 and 6. Find corresponding eigenvectors.

Hence find a matrix $\mathbf { P }$ such that $\mathbf { A } = \mathbf { P D P } \mathbf { P } ^ { - 1 }$, where $\mathbf { D }$ is a diagonal matrix which is to be determined.

The matrix $\mathbf { B }$ is such that $\mathbf { B } = \mathbf { Q A Q } ^ { - 1 }$, where

$$\mathbf { Q } = \left( \begin{array} { r r r } 
4 & 11 & 5 \\
1 & 4 & 2 \\
1 & 2 & 1
\end{array} \right)$$

By using the expression $\mathbf { P D P } ^ { - 1 }$ for $\mathbf { A }$, find the set of eigenvalues and a corresponding set of eigenvectors for $\mathbf { B }$.\\[0pt]
[Question 11 is printed on the next page.]

\hfill \mbox{\textit{CAIE FP1 2015 Q10}}