\begin{definition}[Definition:Characteristic Polynomial of Matrix]
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 $R[x]$:
:$p_M (x) = \operatorname{det}(xI - M)$.
\end{definition}