# Definition:Characteristic Polynomial of Matrix

## Definition

Let $R$ be a commutative ring with unity.

Let $M$ be a square matrix over $R$ of order $n > 0$.

Let $I_n$ be the $n\times n$ identity matrix.

Let $R[x]$ be the polynomial ring in one variable over $R$.

The characteristic polynomial of $M$ is the determinant of a matrix over $R[x]$:

$p_M (x) = \operatorname{det}(xI - M)$.