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: Linear Algebra ABC Proof

  1. Feb 8, 2009 #1
    1. The problem statement, all variables and given/known data
    Prove that if A,B,and C are square matrices and ABC = I, then B is invertible and B^-1 = CA.

    3. The attempt at a solution

    [tex]ABC = I[/tex]
    [tex]CABC = CI[/tex]
    [tex]CABC = C[/tex]
    [tex]CABCA = CA[/tex]

    so we have these two things:
    [tex](CAB)CA = CA[/tex]
    [tex]CA(BCA) = CA[/tex]

    so I thought that since CA times CAB = CA then CAB = I, and same for BCA. But, that is only true if the matrix is invertible, and the problem doesn't say whether C and A are invertible. Any suggestions? Thanks.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Feb 8, 2009 #2

    Mark44

    Staff: Mentor

    If M and N are square matrices, and MN = I, then both M and N are invertible.

    You're given that ABC = I, so A(BC) = I, and (AB)C = I, which says that A, BC, AB, and C are all invertible, and that A^(-1) = BC, and so on.

    Your last equation is CA(BCA) = CA, which suggests to me that BCA = I, or that B(CA) = I. What's the relationship between B and CA?
     
  4. Feb 9, 2009 #3
    Is an if you multiply invertible matrices together does it always yield an invertible matrix?

    If so then CA(BCA) = CA implies that B(CA) = I, which means that B^-1=CA.
     
  5. Feb 9, 2009 #4
    Yes because det(AB) = det(A)*det(B) and if det(AB) is non-zero, then neither A nor B can have zero determinant, i.e., they must both be invertible. [if we're dealing with real or complex valued matrices at least]
     
  6. Feb 9, 2009 #5
    Yes we are. Alright then, thank you both.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook