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Question about invertible matrices

  1. Feb 9, 2009 #1
    1. The problem statement, all variables and given/known data
    A, B, I-Y^-1, X, Y are n x n invertible matrices

    (A(I-Y^-1))^-1 = YB

    Solve for Y

    2. Relevant equations



    3. The attempt at a solution

    (I - Y^-1)^-1A^-1 = YB
    (I - Y^-1)^-1A^-1B^-1 = Y
    A^-1B^-1 = (I - Y^-1)Y
    A^-1B^-1 = Y - I
    A^-1B^-1*(A^-1B^-1)^-1 = Y - I(A^-1B^-1)^-1
    I = Y - (A^-1B^-1)^-1
    Y^-1*I = Y^-1*Y - (A^-1B^-1)^-1
    Y^-1 = I - (A^-1B^-1)^-1
    Inverse both sides
    Y = (I - (A^-1B^-1)^-1)^-1
    or is it
    Y = I^-1 - A^-1B^-1
    Y = I - A^-1B^-1

    Is this correct?
     
  2. jcsd
  3. Feb 9, 2009 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    You can solve for Y from here immediately:
    A^-1B^-2+ I= Y

    Don't you mean = (Y- I)(A^-1B^-1)^-1

    This is wrong now.

     
  4. Feb 9, 2009 #3
    Didn't know you could move -I to the other side like normal algebra.

    Is the B^-2 a typo? Cause I don't know what that's supposed to represent.
     
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