1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    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!

Help with matrix proof

  1. Sep 23, 2012 #1
    the matrices A and B are
    invertible symmetric matrices and AB = BA.
    Show that A*B^-1 is symmetric

    =A^T * (B^-1)^T
    =A^T * (B^T)^-1

    Since A and B are symmetric

    Is this right? Is (B^-1)^T = (B^T)^-1?
  2. jcsd
  3. Sep 23, 2012 #2


    User Avatar
    Science Advisor

    Yes, that is true, but "(A*B^-1)^T= A^T*(B^-1)^T" isn't.
    Rather, (A*B^{-1})^T= (B^{-1})^T*B^T.
  4. Sep 23, 2012 #3
    Oh I memorized the identity wrong. (AB^T)=B^t*A^T
  5. Sep 23, 2012 #4
    The solution in the book first proves
    IF AB=BA, then B^-1 * A *B=A, so B^-1*A=AB^-1

    For the last type, the "B^-1*A=AB^-1", part how did they go from B^-1 * A *B=A to

    Did they divide both sides by B?
  6. Sep 23, 2012 #5


    User Avatar
    Science Advisor
    Homework Helper

    They multiplied both sides on the right by B^(-1). Talking about 'dividing' matrices by B is ambiguous. You can 'divide' on the left or the right.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook