# Definition of elementary matrix

Put content here.**Definition.** An *elementary matrix* is a matrix obtained by performing a single elementary row operation on the identity matrix $I_n$.
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There are three types of elementary matrices corresponding to the three types of row operations:
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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 $I_n$.
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**Example.** For $n=3$, the elementary matrix that swaps rows 1 and 2 is:
$$E_{12} = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$

# Parents

* Elementary matrices