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Proof: given that AB=I,BA=I

  1. Feb 16, 2010 #1
    Let A and B be 2x2 matrices s.t. AB=I . Then how can I prove that BA=I?


    I assumed that there must exist some sequence of elementary row operations which carries B into I, and I denoted this sequence by the matrix A.

    But here, I realized there's some pieces that I' m missing, which I colored red.

    How can I explain it ? or is the way of proving this statement even valid?

    Somebody help me plz!!!!
     
  2. jcsd
  3. Feb 16, 2010 #2

    radou

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    If, for example, A is regular, then its inverse A^-1 = B and hence AB = BA = I. But, in general AB does not equal BA.
     
  4. Feb 16, 2010 #3

    Fredrik

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    This looks like homework, so it should probably be in the homework forum.

    I'll give you some hints:

    1. Can you prove this if you know that at least one of the matrices is invertible?
    2. Can you prove that at least one of them must be invertible?
     
  5. Feb 16, 2010 #4

    Redbelly98

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    Moderator's note:
    Homework assignments or textbook style exercises for which you are seeking assistance are to be posted in the appropriate forum in our https://www.physicsforums.com/forumdisplay.php?f=152" area. This should be done whether the problem is part of one's assigned coursework or just independent study.
     
    Last edited by a moderator: Apr 24, 2017
  6. Feb 21, 2010 #5
  7. Feb 21, 2010 #6

    Fredrik

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    It's really easy to do it the way I suggested, so I strongly suggesting that cocobaby try to do it that way instead of trying to find the proof in a book.
     
  8. Feb 22, 2010 #7

    HallsofIvy

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    Together with Fredrick's suggestion, use the fact that det(AB)= det(A)det(B)= det(I)= 1.
     
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