How to Prove det(xIm - AB) = xm-ndet(xIn - BA)?

[brru25]

1. The problem statement

For integers m >= n,

Prove det(xI_m - AB) = x^m-ndet(xI_n - BA) for any x in R.

Homework Equations

A is an m x n matrix
B is an n x m matrix

The Attempt at a Solution

I tried working out the characteristic polynomials by hand but it just seems too tedious for a nice proof. I know that each x is an eigenvalue of AB but after that I'm stumped.

 

[miqbal]

Sylvester's determinant theorem covers this: http://en.wikipedia.org/wiki/Determinant#Sylvester.27s_determinant_theorem

A link to the proof is in the citation.