What is the relation between two row equivalent matrices? My question is very simple. 
Suppose $[T]_{\mathfrak B}$ is the matrix of the linear transformation $T:V\to V$ in the basis $\mathfrak B$. 

If a matrix $A$ is row equivalent to $[T]_{\mathfrak B}$, what is the relation between these two matrices? 

Can we know something about the linear transformation $T$ just looking to the matrix $A$?

Note that Elementary row operation of a matrix $A$ means multiplication of $A$ by an elementary matrix from the left.
A matrix $E$ is called elementary if it can be obtained from the identity matrix $I$ by a single row operation.
Since $B$ is obtained from $A$ by elementary row operations so $B$ will have the form $B=PA$ where $P=E_kE_{k-1}\ldots E_1$ where $E_i$ denotes an elementary row operation and 
$k=$Number of operations needed to obtain $B$ from $A$. 
