proof of Cayley-Hamilton theorem in a commutative ring

Let $R$ be a commutative ring with identity and let $A$ be an order $n$ matrixwith 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:

A^{\\sigma}=\\left[{0\\atop 1}\\quad{0\\atop 5}\\right]+\\left[{2\\atop 1}\\quad{0\\atop0%}\\right]x+\\left[{1\\atop 0}\\quad{7\\atop 0}\\right]x^{2}.

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 $A\\in M_{n}(R)$ andconsider the characteristic polynomial of $A$ : $p_{A}(x)=\\det(xI-A)$ , which is a monicpolynomial 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 factorof $p_{A}(x)$ . 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 $c I$ , for $c\\in R$ ,so they commute with $A$ . Therefore right and left hand values are the same.

References

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