> [!Definition] Definition (Elementary row operation) > Given a [[Matrix|matrix]], an **elementary row operation** (or **ERO**) is one of the following: > 1. [[Matrix Row|rows]] may be swapped denotede $S_{ij}$ > 2. A row can be multiplied by a non-zero scalar; > 3. A multiple of one row may be added to a second row. > > > [!info] Language > We introduce some notation to help us talk about elementary row operations. Note that this is not standard notation, however it is convenient to have. > 1. Let $S_{ij}$ denote the elementary row operation which swaps rows $i$ and $j$. > 2. Let $M_{i}(\lambda)$ denote the elementary row operation which multiples row $i$ by $\lambda$. > 3. Let $A_{ij}(\lambda)$, where $i \neq j$, denote the elementary row operation which adds $\lambda$ times row $i$ to row $j$. >