Elementary Row Operations are Invertible by other Elementary Row Operations

Last updated Nov 1, 2022

Let $F$ be a Field, let $m, n \in \mathbb{N}$, $e \in \text{Row}F^{m \times \cdot}$. Then there exists $e’ \in \text{Row}F^{m \times \cdot}$ of the same type so that $\forall A \in F^{m \times \cdot}$, $$e’(e(A)) = A = e(e’(A))$$

We will prove this case by case for the 3 different kinds of Elementary Row Operations.

Let $e$ be the Elementary Row Operation that scales row $r \in [m]$ by $c \in F$. Observe scaling row $r \in [m]$ by $c^{-1} \in F$ is also an Elementary Row Operation. Call it $e’$. Then $$e’(e(A)){ij} = \begin{cases}A{ij} & \text{if }i \neq r \\ c^{-1}c A_{rj} & \text{if } i =r \end{cases} = A_{ij},$$ so $e’(e(A)) = A$. Because $c = (c^{-1})^{-1}$, the same argument tells us $e(e’(A)) = A$. $\checkmark$

Let $e$ be the Elementary Row Operation that adds row $r \in [m]$ scaled by $c \in F$ to row $s \in [m]$. Observe that adding row $r \in [m]$ scaled by $-c \in F$ is another Elementary Row Operation. Call it $e’$. Then $$e’(e(A)){ij}= \begin{cases} A{ij} & \text{if } i \neq r \\ (A_{sj} + cA_{rj}) - cA_{rj} & \text{if } i = s \\ \end{cases} = A_{ij},$$ so $e’(e(A)) = A$. Since $c = -(-c)$, the same argument tells us $e(e’(A)) = A$. $\checkmark$

Let $e$ be the Elementary Row Operation that swaps row $r \in [m]$ with row $s \in [m]$. Then if we apply $e$ to itself, we get $$e(e(A)){ij} = \begin{cases} A{ij} & \text{if } i \neq r, i \neq s \\ A_{rj} & \text{if } i = r \\ A_{sj} & \text{if } i = s \\ \end{cases} = A_{ij}$$ so $e(e(A)) = A$. $\checkmark$

$\blacksquare$

• Elementary Row Operation
• Matrix
• Natural Numbers