1. Limited time only! Sign up for a free 30min personal 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!

Homework Help: Disprove that AB-BA = I

  1. Jan 6, 2012 #1
    The task is to prove that for no two matrices A and B, A*B - B*A = I, where I is the identity matrix.
    I tried multiplying by the inverses of A or B, but that doesn't seem to lead to a more manageable form. The only way I see this could be done is by writing down all n*n (assuming n by n matrices) linear equations. It's easy to do when n = 2, but the same contradiction may not be as obvious for higher n.
    I hope there is a more intelligent way to go about this.
  2. jcsd
  3. Jan 6, 2012 #2


    User Avatar
    Science Advisor

    What do you know about determinants?
  4. Jan 6, 2012 #3
    I know that det(AB) = det(BA), but I don't know what are the properties when subtraction is involved. Except for the case when only one line is different.
  5. Jan 6, 2012 #4


    User Avatar

    Staff: Mentor

    Determinant is just a number, isn't it?
  6. Jan 6, 2012 #5
    What I mean is that I don't know what is det(AB-BA) even if I do know det(AB) and det(BA).
    I'm looking at Sylvester's determinant theorem which looks related, but I still don't see a solution. Now I need to prove that for no M, det(M+I) = det(M)[STRIKE], at least when M = AB..[/STRIKE] (now that I think about it, there is probably no matrix that can't be written as a product of two others, is there?)
  7. Jan 6, 2012 #6


    User Avatar
    Science Advisor
    Homework Helper

    Try taking the trace.
  8. Jan 6, 2012 #7
    So tr(AB-BA) = 0 ? Great. Thanks.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook