proof of Cayley-Hamilton theorem in a commutative ring

Let R be a commutative ring with identityPlanetmathPlanetmath and let A be an order n matrix with elements from R[x]. For example, if A is (x2+2x7x2x+15)

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


In this way we have a mapping AAσ which is an isomorphismPlanetmathPlanetmathPlanetmathPlanetmath of the rings Mn(R[x]) and Mn(R)[x].

Now let AMn(R) and consider the characteristic polynomialMathworldPlanetmathPlanetmath of A: pA(x)=det(xI-A), which is a monic polynomialMathworldPlanetmath of degree n with coefficients in R. Using a property of the adjugate matrix we have


Now view this as an equation in Mn(R)[x]. It says that xI-A is a left factor of pA(x). So by the factor theorem, the left hand value of pA(x) at x=A is 0. The coefficients of pA(x) have the form cI, for cR, so they commute with A. Therefore right and left hand values are the same.


  • 1 Malcom F. Smiley. AlgebraMathworldPlanetmath of Matrices. Allyn and Bacon, Inc., 1965. Boston, Mass.
Title proof of Cayley-Hamilton theoremMathworldPlanetmath 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