Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Invertible Matrices

  1. Mar 29, 2007 #1

    daniel_i_l

    User Avatar
    Gold Member

    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. jcsd
  3. Mar 29, 2007 #2

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    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.)
     
  4. Mar 29, 2007 #3

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    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).
     
  5. Mar 29, 2007 #4

    daniel_i_l

    User Avatar
    Gold Member

    Thanks a lot! now i can go to sleep...
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook