# Thread: Proof of equivalent matrices

1. ## 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.

2. 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.

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

3. Thank you very much!