Matrix Representation of Elementary Row Operations

Last updated Nov 1, 2022

Let $F$ be a Field and let $m \in \mathbb{N}$.. All Elementary Row Operations of $\text{Row} F^{m \times \cdot}$ can be represented as elements of $F^{m \times m}$. The representations are

1. Row scaling row $r \in [m]$ by $c \in F$can be represented as $$R_{ij} = \begin{cases} 1 & \text{if } i = j \neq r \\ c & \text{if } i=j=r \\ 0 & \text{otherwise} \end{cases}$$
2. Row adding row $r \in [m]$ scaled by $c \in F$ to row $s \in [m]$ can be represented as $$R_{ij} = \begin{cases} 1 & \text{if } i = j \\ c & \text{if } i=s, j=r \\ 0 & \text{otherwise} \end{cases}$$
3. Row swapping row $r \in [m]$ with row $s \in [m]$ can be represented as $$R_{ij} = \begin{cases} 1 & \text{if } i = j \neq r, s \\ 1 & \text{if } i=s, j=r \\ 1 & \text{if } i=r, j=s \\ 0 & \text{otherwise} \end{cases}$$ Application of these Elementary Row Operations can be represented by Matrix Multiplication.

TODO, the proof is simple, just apply each one and show they give the desired result.