Elementary Column Matrix is Invertible
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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.
$\blacksquare$