1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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...
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Invertible Matrices
  1. Invertible Matrices (Replies: 2)

  2. Invertible matrices (Replies: 3)

Loading...