Problem:
Find the eigenvalues of the following matrix:
$\left(
\begin{array}{ccc}
 -3 & -2 & 5 \\
 -1 & 0 & 7 \\
 -9 & 2 & -9 \\
\end{array}
\right)$.
Answer:
\begin{array}{l}
 
\begin{array}{l}
 \text{Find all the eigenvalues of the matrix }M: \\
 M=\left(
\begin{array}{ccc}
 -3 & -2 & 5 \\
 -1 & 0 & 7 \\
 -9 & 2 & -9 \\
\end{array}
\right) \\
\end{array}
 \\
\hline
 
\begin{array}{l}
 \text{Find }\lambda \in \mathbb{C} \text{such }\text{that }M v=\lambda  v \text{for }\text{some }\text{nonzero }\text{vector }v: \\
 M v=\lambda  v \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Rewrite }\text{the }\text{equation }M v=\lambda  v \text{as }(M-\mathbb{I} \lambda ) v=0: \\
 (M-\mathbb{I} \lambda ) v=0 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Find }\text{all }\lambda  \text{such }\text{that }| M-\mathbb{I} \lambda | =0: \\
 | M-\mathbb{I} \lambda | =0 \\
\end{array}
 \\
 
\begin{array}{l}
 
\begin{array}{lll}
 M-\mathbb{I} \lambda  & = & \fbox{$\left(
\begin{array}{ccc}
 -3 & -2 & 5 \\
 -1 & 0 & 7 \\
 -9 & 2 & -9 \\
\end{array}
\right)$}-\fbox{$\left(
\begin{array}{ccc}
 1 & 0 & 0 \\
 0 & 1 & 0 \\
 0 & 0 & 1 \\
\end{array}
\right)$} \lambda  \\
 \text{} & = & \left(
\begin{array}{ccc}
 -\lambda -3 & -2 & 5 \\
 -1 & -\lambda  & 7 \\
 -9 & 2 & -\lambda -9 \\
\end{array}
\right) \\
\end{array}
 \\
 \left| 
\begin{array}{ccc}
 -\lambda -3 & -2 & 5 \\
 -1 & -\lambda  & 7 \\
 -9 & 2 & -\lambda -9 \\
\end{array}
\right| =0 \\
\end{array}
 \\
 
\begin{array}{l}
 \left| 
\begin{array}{ccc}
 -\lambda -3 & -2 & 5 \\
 -1 & -\lambda  & 7 \\
 -9 & 2 & -\lambda -9 \\
\end{array}
\right| =-\lambda ^3-12 \lambda ^2-56 \lambda +176: \\
 -\lambda ^3-12 \lambda ^2-56 \lambda +176=0 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Solving }\text{the }\text{equation }-\lambda ^3-12 \lambda ^2-56 \lambda +176=0 \text{for }\lambda  \text{yields }\text{the }\text{following }\text{eigenvalues}: \\
 \fbox{$
\begin{array}{ll}
 \text{Answer:} &  \\
 \text{} & \lambda =2.06824\text{ or }\lambda =-7.03412-5.96805 i\text{ or }\lambda =-7.03412+5.96805 i \\
\end{array}
$} \\
\end{array}
 \\
\end{array}

{'source': 'AMPS', 'file': 'amps/mathematica/linear_algebra/eigenvalues_w_steps/2004.txt'}