Elementary Column Matrix is Invertible

Theorem

Let $\mathbf E$ be an elementary column matrix.

Then $\mathbf E$ is invertible.

Proof

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

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

then:

the inverse of $e$ corresponds to an elementary column matrix which is the inverse of $\mathbf E$.

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

$\blacksquare$

Also see

• Elementary Row Matrix is Invertible