characteristic polynomial

$p_A(\lambda)=\det(A-\lambda I)$ , a degree- $n$ polynomial whose roots are $A$ 's eigenvalues.

For an $n\times n$ matrix $A$ , $p_A(\lambda)=\det(A-\lambda I)$ is the characteristic polynomial — always degree exactly $n$ in $\lambda$ , with leading term $(-1)^n\lambda^n$ coming from the diagonal product $(a_{11}-\lambda)\cdots(a_{nn}-\lambda)$ . $\lambda_0$ is a root of $p_A$ exactly when $\lambda_0$ is an eigenvalue of $A$ .