Elementary Column Matrix is Nonsingular

Theorem

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

Then $\mathbf E$ is nonsingular.


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 nonsingular.

$\blacksquare$


Also see

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