# Prove that a square matrix is nonsingular if and only if it is row-equivalent to an identity matrix.

I know that the identity matrix would look something like $\begin{pmatrix} 1 & 0 & 0 \\0 &1 & 0 \\ 0 & 0 &1 \end{pmatrix}$, which I know is non singular, but I am not sure how to go about proving that a square matrix is nonsingular if and only if it is row-equivalent to an identity matrix.

• Maybe you can start by considering the system $Ax = b$. Matrix being nonsingular $\Longleftrightarrow$ the system has unique solution $\Longleftrightarrow$ row-equivalent to an identity matrix – 3x89g2 Sep 16 '16 at 16:48

So for any $A$ there is $B$, invertible so that $BA$ is reduced row echelon. If $BA$ is the identity then $BA=I$ and $B$ is the inverse for $A$. If $BA$ is not the identity then it has a zero row and thus $BA$ cannot be invertible and since $B$ is invertible $A$ cannot be invertible.