Elementary Row Matrix is Invertible

Theorem
Let $\mathbf E$ be an elementary row matrix.

Then $\mathbf E$ is invertible.

Proof
From Elementary Row Matrix for Inverse of Elementary Row Operation is Inverse it is demonstrated that:


 * if $\mathbf E$ is the elementary row matrix corresponding to an elementary row operation $e$

then:
 * the inverse of $e$ corresponds to an elementary row matrix which is the inverse of $\mathbf E$.

So as $\mathbf E$ has an inverse, a priori it is invertible.

Also see

 * Elementary Column Matrix is Invertible