Definition of elementary matrix

Definition. An elementary matrix is a matrix obtained by performing a single elementary row operation on the identity matrix $I_n$.

There are three types of elementary matrices corresponding to the three types of row operations:

1. Row interchange: $E_{ij}$ swaps rows $i$ and $j$ of $I_n$.
2. Row scaling: $E_i(c)$ multiplies row $i$ of $I_n$ by a nonzero scalar $c$.
3. Row replacement: $E_{ij}(c)$ adds $c$ times row $j$ to row $i$ of $I_n$.

Example. For $n=3$, the elementary matrix that swaps rows 1 and 2 is:
$$E_{12} = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$