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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Originally Posted by Brokescholar

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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I do it for matrices to give you an idea. If is equivalent to then it means for an invertible matrix . This means . Thus, is equivalent to .

Thank you very much!