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Easy Linear Algebra Question

  1. Sep 26, 2007 #1
    The question is: Suppose one has n×n square matrices X, Y and Z such that
    XY = I and Y Z = I. Show that it follows that X = Z.

    Now if this were XY and ZY, I would just say that:

    XY=ZY -> XY-ZY=0 ->Y(X-Z)=0 -> X-Z=0 -> X=Z.

    I am wondering that since the Y is on different sides of the Z and X this still holds? Thanks!
     
  2. jcsd
  3. Sep 26, 2007 #2
    I don't think you can just pull the Y out like that.

    If I recall correctly, if XY = I, then doesn't YX = I aswell? If this is true perhaps you could encorporate it into your proof somehow.
     
  4. Sep 26, 2007 #3
    Actually I think I just figured it out.
    X^-1 Y =X^-1 I -> Y=X^-1

    then sub that in for Y in the other equation
    X^-1 Z = I
    XX^-1 Z = X
    Z=X

    Thanks for your help!
     
  5. Sep 26, 2007 #4

    mathwonk

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    it goes like this: XY = I so XYZ = IZ = Z.

    but YZ = I too, so XYZ = XI = X. so X = XYZ = Z.

    the problem with your argument is that there is no X^-1 unless yoiu prove this problem.

    but if yu are careful, yu can use the fact that X = the left inverse of Y and Z = the right inverse of Y.
     
  6. Sep 26, 2007 #5
    Oh, I forgot the inverses don't always exist...it has been a bit since I have done and linear algebra work. Your solution makes perfect sense, thanks!
     
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