Invertible Matrices

  • Thread starter daniel_i_l
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  • #1
daniel_i_l
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Homework Statement


Q: If A and B are both nxn matrices and AB-I is invertable then prove that BA-I is also invertable.


Homework Equations


if A is invertible iff |A|<>0


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.
 

Answers and Replies

  • #2
matt grime
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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
Dick
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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
daniel_i_l
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Thanks a lot! now i can go to sleep...
 

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