Let $R$ be a commutative ring with identity and let $A$ be an order $n$ matrix with elements from $R[x]$ . For example, if $A$ is $\begin{pmatrix} x^2+2x & 7x^2 \\ x+1 & 5 \end{pmatrix}$

then we can also associate with $A$ the following polynomial having matrix coefficents:

In this way we have a mapping $A\longrightarrow A^\sigma$ which is an isomorphism of the rings $M_{n}(R[x])$ and $M_{n}(R)[x]$ .

Now let $A \in M_{n}(R)$ and consider the characteristic polynomial of $A$ : $p_{A}(x) = \det(xI - A)$ , which is a monic polynomial of degree $n$ with coefficients in $R$ . Using a property of the adjugate matrix we have $$(xI-A)\operatorname{adj}(xI-A) = p_{A}(x)I.$$ Now view this as an equation in $M_{n}(R)[x]$ . It says that $xI-A$ is a left factor of $p_{A}(x)$ . So by the factor theorem, the left hand value of $p_{A}(x)$ at $x=A$ is 0. The coefficients of $p_{A}(x)$ have the form $cI$ , for $c\in R$ , so they commute with $A$ . Therefore right and left hand values are the same.

Bibliography

1

Malcom F. Smiley. Algebra of Matrices. Allyn and Bacon, Inc., 1965. Boston, Mass.