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 fortiori it is invertible.

$\blacksquare$

Also see

• Elementary Column Matrix is Invertible

Sources

• 1995: John B. Fraleigh and Raymond A. Beauregard: Linear Algebra (3rd ed.) $\S 1.5$