Find the eigenvalues and corresponding eigenvectors for the matrix \(\mathbf { A }\), where
$$\mathbf { A } = \left( \begin{array} { l l }
6 & - 3
4 & - 1
\end{array} \right)$$
Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } = \mathbf { P D P } ^ { - 1 }\).
The \(3 \times 3\) matrix \(\mathbf { B }\) has characteristic equation
$$\lambda ^ { 3 } - 4 \lambda ^ { 2 } - 3 \lambda - 10 = 0$$
Show that 5 is an eigenvalue of \(\mathbf { B }\). Show that \(\mathbf { B }\) has no other real eigenvalues.
An eigenvector corresponding to the eigenvalue 5 is \(\left( \begin{array} { r } - 2 1 4 \end{array} \right)\).
Evaluate \(\mathbf { B } \left( \begin{array} { r } - 2 1 4 \end{array} \right)\) and \(\mathbf { B } ^ { 2 } \left( \begin{array} { r } 4 - 2 - 8 \end{array} \right)\).
Solve the equation \(\mathbf { B } \left( \begin{array} { l } x y z \end{array} \right) = \left( \begin{array} { r } - 20 10 40 \end{array} \right)\) for \(x , y , z\).
Show that \(\mathbf { B } ^ { 4 } = 19 \mathbf { B } ^ { 2 } + 22 \mathbf { B } + 40 \mathbf { I }\).