> [!Proposition] Proposition (Elementary Matrices for Row Operations) > Let $A\in \text{Mat}_{mn}(\mathbb{R})$ be a [[Real Matrices|real matrix]]. Applying any of the [[Elementary Row Operations|elementary row operations]] $S_{IJ},M_{I}(\lambda),A_{IJ}(\lambda)$ is equivalent to [[Real Matrix Product|premultiplying]] $A$ by matrices we also denote by $S_{IJ},M_{I}(\lambda),A_{IJ}(\lambda)$ respectively. These matrices are defined as follows: > >- the $(i, j)$ th entry of $M_I(\lambda)= \begin{cases}1, & i=j, i \neq I \\ \lambda, & i=j=I \\ 0, & \text { otherwise }\end{cases}$ > >- the $(i, j)$ th entry of $A_{I J}(\lambda)= \begin{cases}1, & i=j \\ \lambda, & i=J, j=I \\ 0, & \text { otherwise. }\end{cases}$ > > - the $(i, j)$ th entry of $S_{I J}= \begin{cases}1, & i=j, i \neq I, i \neq J \\ 1, & \{i,j\} = \{I,J\} \\ 0, & \text { otherwise. }\end{cases}$ > > >We call these **elementary matrices**. That is, the elementary matrices are given by applying the corresponding elementary row operation to the [[Real Identity Matrix|identity matrix]] of order $m,$ $I_{m}.$ **Proof**: ...