I got this question on a book and I don't know how to get the correct answer to it. I know that if A and B are row equivalent, they both can be reduced to each other through elementary row operations. And it is the logical conclusion from that point on that only if one of them can be reduced to I matrix can the other also be capable of that. But I don't know how to write it as proof.
Assume that $A$ is non-singular. Then it transforms to the identity matrix $I_n$. Now, start with $B$. First transform $B$ to $A$ and then $A$ to $I$. Hence $B$ can be transformed to $I_n$, so $B$ is non-singular.
If $B$ is non-singular, do the same stuff as above with $A$ and $B$ interchanged.