# 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 $cI$, 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 ProofOfCayleyHamiltonTheoremInACommutativeRing 2013-03-22 16:03:16 2013-03-22 16:03:16 Mathprof (13753) Mathprof (13753) 11 Mathprof (13753) Proof msc 15A18 msc 15A15