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Linear Algerbra. Inverses and Algerbraic Properties of Matrices

  1. Jan 13, 2013 #1
    1. The problem statement, all variables and given/known data
    Assuming that all matrices, A, B, C, and D, are n x n and invertible, solve for D.

    [itex]C^{T}B^{-1}A^{2}BAC^{-1}DA^{-2}B^{T}C^{-2}=C^{T}[/itex]

    2. Relevant equations

    [itex]C^{T}B^{-1}A^{2}BAC^{-1}DA^{-2}B^{T}C^{-2}=C^{T}[/itex]

    3. The attempt at a solution

    I must have missed something in the reading of this section. All I can think of is

    [itex]B^{-1}A^{2}BAC^{-1}DA^{-2}B^{T}C^{-2}=I[/itex]

    but I don't know where to go from there or if it is even the right way to start.
     
  2. jcsd
  3. Jan 13, 2013 #2
    Left-multiply the third equation with B.
     
  4. Jan 13, 2013 #3

    HallsofIvy

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    Staff Emeritus
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    Just "undo" every thing that is done to D, one step at a time. For example if you multiply both sides, on the left, by [itex](C^T)^{-1}[/itex] you get [itex]B^{-1}A^{-2}BAC^{-1}DA^{-2}B^TC^{-2}= (C^T)^{-1}C^T= I[/itex]
    Then multiply on the left by [itex]B[/itex], then on the right by [itex]C^2[/itex], or do both together to get [itex]A^{=2}BAC^{-1}DA^{-2}B^T= BC^2[/itex].
    Continue to "unpeel" D.

    Yes, that was the first step as I showed. Continue in the same way.
     
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