proof of Cayley-Hamilton theorem in a commutative ring

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:

A^{\sigma}=\left[{0\atop 1}\quad{0\atop 5}\right]+\left[{2\atop 1}\quad{0\atop 0% }\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 $A\in M_{n}(R)$ 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 $c I$ , 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.

Title	proof of Cayley-Hamilton theorem in a commutative ring
Canonical name	ProofOfCayleyHamiltonTheoremInACommutativeRing
Date of creation	2013-03-22 16:03:16
Last modified on	2013-03-22 16:03:16
Owner	Mathprof (13753)
Last modified by	Mathprof (13753)
Numerical id	11
Author	Mathprof (13753)
Entry type	Proof
Classification	msc 15A18
Classification	msc 15A15