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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.