---
tag: 0M79HRJCRR
kind: definition
name: Elementary matrix
msc: [15A23]
defines: 01REXW3WWE
uses: [06DW9BAWB1, 0DE5R5EB4D]
sources:
  - url: https://en.wikipedia.org/wiki/Elementary_matrix
    locator: lead section
---

## Definition

Let $F$ be a field and let $n \ge 1$ be an integer.
An $n \times n$ *elementary matrix* over $F$ is a matrix that results from the [[identity matrix|06DW9BAWB1]] $I_n$ by one [[elementary row operation|04XWH28452]].
There are three types:

1. $P_{ij}$, for $i \ne j$: exchange rows $i$ and $j$ of $I_n$.
2. $D_i(c)$, for $c \ne 0$: multiply row $i$ of $I_n$ by $c$.
3. $T_{ij}(c)$, for $i \ne j$ and $c \in F$: add $c$ times row $j$ of $I_n$ to row $i$.

## Remarks

For each $n \times p$ matrix $A$, the product $EA$ is the result of the same row operation on $A$.
The product $AE$ performs the matching column operation on $A$.
Each elementary matrix is invertible, and its inverse is an elementary matrix of the same type.
Each invertible matrix over a field is a product of elementary matrices.

## Explanations

### L1 human

An elementary matrix stores one row operation as a matrix.
To do the operation on $A$, we multiply $A$ on the left by this matrix.
Then a whole elimination becomes a product of elementary matrices.