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Math Help - Proof of equivalent matrices

  1. #1
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    Proof of equivalent matrices

    Let A and B be m x n matrices. Show that A is equivalent to B if and only if AT is equivalent to BT.

    So far I've got this. I'm not totally sure if it's correct or if I'm missing something major. Thanks!


    AT = PBTQ for some nonsingular matrices P and Q.

    Taking the transpose of both sides I get

    A = PTBQT

    (PT)-1A(QT)-1 = B

    By definition A is equiv. to B because we can get B from A by a sequence of elem. row/column operations.


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  2. #2
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    Quote Originally Posted by Brokescholar View Post
    Let A and B be m x n matrices. Show that A is equivalent to B if and only if AT is equivalent to BT.

    So far I've got this. I'm not totally sure if it's correct or if I'm missing something major. Thanks!


    AT = PBTQ for some nonsingular matrices P and Q.

    Taking the transpose of both sides I get

    A = PTBQT

    (PT)-1A(QT)-1 = B

    By definition A is equiv. to B because we can get B from A by a sequence of elem. row/column operations.


    I do it for n\times n matrices to give you an idea. If A is equivalent to B then it means A=MBM^{-1} for an invertible matrix M. This means A^{T} = (MBM^{-1})^T = (M^T)^{-1} B^T M^T. Thus, A^T is equivalent to B^T.
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  3. #3
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    Thank you very much!
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