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Homework Help: 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*B^-1)^T
    =A^T * (B^-1)^T
    =A^T * (B^T)^-1

    Since A and B are symmetric
    =A*B^-1


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

    HallsofIvy

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    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
    B^-1*A=AB^-1.

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

    Dick

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    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.
     
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