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Showing a matrix is invertible

  1. Feb 19, 2014 #1
    1. The problem statement, all variables and given/known data

    let A and B be nXn invertible matrices. Show that (A.B)T is invertible by finding it's inverse

    2. Relevant equations



    3. The attempt at a solution

    (A.B)T=BT.AT
     
    Last edited: Feb 19, 2014
  2. jcsd
  3. Feb 19, 2014 #2

    CAF123

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    Gold Member

    You are looking for a matrix ##C## such that ##(AB)^{T}C = C(AB)^T = I_{n}##. The question does not say that A and B are invertible, however given that the inverse of (AB)T exists you can show that they are indeed invertible.
     
  4. Feb 19, 2014 #3
    Sorry I left out the fact that A and B are invertible.
     
  5. Feb 19, 2014 #4

    HallsofIvy

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    You have already said that [itex](AB)^T= B^TA^T[/itex]. Okay, what will get rid of that [itex]B^T[/itex] and then what will get rid of that [itex]A^T[/itex]

    (Of course, you know that a matrix is invertible if and only if its determinant is not 0? And that the determinant of [itex]X^T[/itex] is the same as the determinant of [itex]X[/itex]?)
     
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