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 $A$?


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

Yes, we can. For instance, the rank of $A$ is equal to that of $[T]_{\mathcal{B}}$, which would give you the information about where $T$ is invertible or not. 
