Characteristic polynomial

The characteristic polynomial of a matrix A is the polynomial satisfies the equation

$|\lambda I_n-A|=0$

The roots of this function will be the eigenvalues of the matrix.

Example

Given the matrix

$A=\begin{bmatrix}3&0\\-1&-1\end{bmatrix}$

The characteristic polynomial will be

$\begin{vmatrix}\lambda I-\begin{bmatrix}3&0\\-1&-1\end{bmatrix}\end{vmatrix}=\begin{vmatrix}\lambda\begin{bmatrix}1&0\\0&1\end{bmatrix}-\begin{bmatrix}3&0\\-1&-1\end{bmatrix}\end{vmatrix}=0$

$\begin{vmatrix}\lambda-3&0\\1&\lambda+1\end{vmatrix}=(\lambda-3)(\lambda+1)-(0)(1)=(\lambda-3)(\lambda+1)=0$

The eigenvalues of A will be -1 and 3.