
#1
Oct2809, 10:09 AM

P: 29

1. The problem statement
For integers m >= n, Prove det(xI_{m}  AB) = x^{mn}det(xI_{n}  BA) for any x in R. 2. Relevant equations A is an m x n matrix B is an n x m matrix 3. 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. 



#2
Oct2809, 02:22 PM

P: 28

Sylvester's determinant theorem covers this: http://en.wikipedia.org/wiki/Determi...minant_theorem
A link to the proof is in the citation. 


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