Elementary row operations

Definition

Let $F$ be a field and let $A$ be an $m \times n$ matrix over $F$, with rows $R_1, \dots, R_m$. An elementary row operation on $A$ is one of these three operations:

1. Row switching: exchange $R_i$ and $R_j$ for some $i \ne j$.
2. Row scaling: replace $R_i$ by $c R_i$ for some $c \in F$ with $c \ne 0$.
3. Row addition: replace $R_i$ by $R_i + c R_j$ for some $i \ne j$ and some $c \in F$.

Remarks

Each elementary row operation has an inverse operation of the same type. Each elementary row operation on $A$ is the same as the product $EA$ for an elementary matrix $E$. Elementary row operations on an augmented matrix do not change the solution set of the system. Elementary column operations have the same definition with columns in place of rows.

Explanations

L1 human

The three row operations are the moves that we make when we solve equations by hand. We can swap two equations, multiply an equation by a nonzero number, or add a multiple of one equation to another. None of these moves gains or loses a solution, and each move can be undone.