Every nonsingular $m\times m$ matrix is row equivalent to identity matrix $I_m$

Solution 1:

Matrix equivalence can be completely characterized in terms of a factorization: $A \sim B$ if and only if there exist nonsingular $P,Q$ of appropriate order so that $A=PBQ$. The matrices $P$ and $Q$ are the products of elementary row and column operation matrices respectively.

Now if $A$ is a nonsingular $m \times m$ matrix, we have $I=A^{-1}A=A^{-1}AI$, with $A^{-1}$ and $I$ nonsingular...so $A \sim I$.

Edit: Sorry I see you refer to row equivalence specifically. There is a similar characterization: $A \stackrel{R}{\sim} B$ iff there exists nonsingular $P$ so that $A=PB$ where $P$ is a product of elementary matrices representing elementary row operations. You can fill in the details....