8 Find the eigenvalues and corresponding eigenvectors of the matrix \(\mathbf { A } = \left( \begin{array} { r r r } 4 & - 1 & 1 \\ - 1 & 0 & - 3 \\ 1 & - 3 & 0 \end{array} \right)\).
Find a non-singular matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { 5 } = \mathbf { P D P } ^ { - 1 }\).
8 Find the eigenvalues and corresponding eigenvectors of the matrix $\mathbf { A } = \left( \begin{array} { r r r } 4 & - 1 & 1 \\ - 1 & 0 & - 3 \\ 1 & - 3 & 0 \end{array} \right)$.
Find a non-singular matrix $\mathbf { P }$ and a diagonal matrix $\mathbf { D }$ such that $\mathbf { A } ^ { 5 } = \mathbf { P D P } ^ { - 1 }$.
\hfill \mbox{\textit{CAIE FP1 2011 Q8}}