Homework Help: Invertible Matrices

1. Mar 29, 2007

daniel_i_l

1. The problem statement, all variables and given/known data
Q: If A and B are both nxn matrices and AB-I is invertable then prove that BA-I is also invertable.

2. Relevant equations
if A is invertible iff |A|<>0

3. The attempt at a solution
I've been thinking about this for over an hour I've only managed to prove it if either A or B are invertable. because if let's say A is invertable then:
|AB-I|<>0 => |AB-I||A|<>0 => |ABA-A|<>0 => |A||BA-I|<>0 => |BA-I|<>0 and so it's invertable. if B is invertable then you do pretty much the same thing on starting on the left side.
But what if they're both singular?
Thanks.

2. Mar 29, 2007

matt grime

X is not invertible if and only if there is a v=/=0 with Xv=0.

Suppose (AB-I)v=0, and see what you can deduce. (I don't promise this works, but is the first thing that springs to mind.)

3. Mar 29, 2007

Dick

You are too hung up on determinants. Try a proof by contradiction. Assume (BA-I) is NOT invertible. Then there is a nonzero vector x such that (BA-I)x=0. Now tell me what is (AB-I)Ax=?. (By playing the same game you did with the determinants).

4. Mar 29, 2007

daniel_i_l

Thanks a lot! now i can go to sleep...