If given a matrix $A$, whose reduced row echelon form you know, can you (without calculating) know the reduced row echelon form of $A^T$? Is there some kind of a connection?
[Math] Reduced row echelon form of $A^T$ if you know the rref of $A$
linear algebra
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Best Answer
If you change one word in the question, that is, from 'row' to 'column' then the question is trivial. Since a matrix is in row echelon form if its transpose is in column echelon form and vice verse. But what you ask is not trivial at all. There is no apparent connection as far as my abilities in Linear Algebra have led me......
Let $ A=\begin{pmatrix} 3 & 4 \\ 3 & 4 \end{pmatrix}$, then CEF of $A$ is $ \begin{pmatrix} 1 & 0 \\ 1 & 0 \end{pmatrix}$ and REF is $\begin{pmatrix} 1 & \frac{4}{3} \\ 0 & 0 \end{pmatrix}$. Where is the connection ? I don't know....
One more thing, if the matrix is invertible, then obviously there is a connection since the echelon forms are the identity.