8 The matrix \(\mathbf { A }\) is given by
$$\mathbf { A } = \left( \begin{array} { r r r }
- 2 & 0 & 0
0 & 7 & 9
4 & 1 & 7
\end{array} \right)$$
- Show that the characteristic equation of \(\mathbf { A }\) is \(\lambda ^ { 3 } - 12 \lambda ^ { 2 } + 12 \lambda + 80 = 0\) and find the eigenvalues of A.
\includegraphics[max width=\textwidth, alt={}, center]{bc601b16-c106-43a2-a2fc-676b5c836096-16_2718_38_106_2010}
\includegraphics[max width=\textwidth, alt={}, center]{bc601b16-c106-43a2-a2fc-676b5c836096-17_2723_33_99_22} - Use the characteristic equation of \(\mathbf { A }\) to show that
$$\mathbf { A } ^ { 4 } = p \mathbf { A } ^ { 2 } + q \mathbf { A } + r \mathbf { I } ,$$
where \(p , q\) and \(r\) are integers to be determined.
- Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(( \mathbf { A } - 3 \mathbf { I } ) ^ { 4 } = \mathbf { P D P } ^ { - 1 }\) .
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