6 The matrix A, where $$\mathbf { A } = \left( \begin{array} { r r r } 1 & 0 & 0 \\ 10 & - 7 & 10 \\ 7 & - 5 & 8 \end{array} \right)$$ has eigenvalues 1 and 3. Find corresponding eigenvectors. It is given that \(\left( \begin{array} { l } 0 \\ 2 \\ 1 \end{array} \right)\) is an eigenvector of \(\mathbf { A }\). Find the corresponding eigenvalue. Find a diagonal matrix \(\mathbf { D }\) and matrices \(\mathbf { P }\) and \(\mathbf { P } ^ { - 1 }\) such that \(\mathbf { P } ^ { - 1 } \mathbf { A P } = \mathbf { D }\).

6 The matrix A, where

$$\mathbf { A } = \left( \begin{array} { r r r } 
1 & 0 & 0 \\
10 & - 7 & 10 \\
7 & - 5 & 8
\end{array} \right)$$

has eigenvalues 1 and 3. Find corresponding eigenvectors.

It is given that $\left( \begin{array} { l } 0 \\ 2 \\ 1 \end{array} \right)$ is an eigenvector of $\mathbf { A }$. Find the corresponding eigenvalue.

Find a diagonal matrix $\mathbf { D }$ and matrices $\mathbf { P }$ and $\mathbf { P } ^ { - 1 }$ such that $\mathbf { P } ^ { - 1 } \mathbf { A P } = \mathbf { D }$.

\hfill \mbox{\textit{CAIE FP1 2015 Q6}}