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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^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


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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

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


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    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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