Problem:
Find the characteristic polynomial of the following matrix:
$\left(
\begin{array}{cc}
 1 & \frac{13}{3} \\
 -10 & -\frac{23}{3} \\
\end{array}
\right)$
Answer:
\begin{array}{l}
 
\begin{array}{l}
 \text{Find the characteristic polynomial of the matrix }M\text{ with respect to the variable }\lambda : \\
 M=\left(
\begin{array}{cc}
 1 & \frac{13}{3} \\
 -10 & -\frac{23}{3} \\
\end{array}
\right) \\
\end{array}
 \\
\hline
 
\begin{array}{l}
 \text{To }\text{find }\text{the }\text{characteristic }\text{polynomial }\text{of }\text{a }\text{matrix, }\text{subtract }\text{a }\text{variable }\text{multiplied }\text{by }\text{the }\text{identity }\text{matrix }\text{and }\text{take }\text{the }\text{determinant}: \\
 | M-\lambda  \mathbb{I}|  \\
\end{array}
 \\
 
\begin{array}{l}
 
\begin{array}{lll}
 | M-\lambda  \mathbb{I}|  & = & \left| 
\begin{array}{cc}
 1 & \frac{13}{3} \\
 -10 & -\frac{23}{3} \\
\end{array}
-\lambda  
\begin{array}{cc}
 1 & 0 \\
 0 & 1 \\
\end{array}
\right|  \\
 \text{} & = & \left| 
\begin{array}{cc}
 1 & \frac{13}{3} \\
 -10 & -\frac{23}{3} \\
\end{array}
-
\begin{array}{cc}
 \lambda  & 0 \\
 0 & \lambda  \\
\end{array}
\right|  \\
\end{array}
 \\
  \text{= }\left| 
\begin{array}{cc}
 1-\lambda  & \frac{13}{3} \\
 -10 & -\lambda -\frac{23}{3} \\
\end{array}
\right|  \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Multiply }\text{along }\text{the }\text{diagonals }\text{and }\text{subtract}: \\
 (1-\lambda ) \left(-\lambda -\frac{23}{3}\right)-\frac{13 (-10)}{3} \\
\end{array}
 \\
 
\begin{array}{l}
 (1-\lambda ) \left(-\lambda -\frac{23}{3}\right)=\lambda ^2+\frac{20 \lambda }{3}-\frac{23}{3}: \\
  \text{= }-\frac{23}{3}+\frac{20 \lambda }{3}+\lambda ^2-\frac{13 (-10)}{3} \\
\end{array}
 \\
 
\begin{array}{l}
 \frac{13}{3} (-10)=-\frac{130}{3}: \\
  \text{= }\left(\lambda ^2+\frac{20 \lambda }{3}-\frac{23}{3}\right)-\left(-\frac{130}{3}\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \left(\lambda ^2+\frac{20 \lambda }{3}-\frac{23}{3}\right)-\left(-\frac{130}{3}\right)=\lambda ^2+\frac{20 \lambda }{3}+\frac{107}{3}: \\
 \fbox{$
\begin{array}{ll}
 \text{Answer:} &  \\
 \text{} &  \text{= }\lambda ^2+\frac{20 \lambda }{3}+\frac{107}{3} \\
\end{array}
$} \\
\end{array}
 \\
\end{array}

{'source': 'AMPS', 'file': 'amps/mathematica/linear_algebra/characteristic_polynomial_w_steps/2568.txt'}