# Definition:Elementary Matrix

## Definition

### Elementary Row Matrix

Let $\mathbf E$ be a unit matrix on which exactly one elementary row operation $e$ has been performed.

Then $\mathbf E$ is called the **elementary row matrix** for the elementary row operation $e$.

### Elementary Column Matrix

Let $\mathbf E$ be a unit matrix on which exactly one elementary column operation $e$ has been performed.

Then $\mathbf E$ is called the **elementary column matrix** for the elementary column operation $e$.

## Also known as

In treatments of linear algebra and matrix theory, it is usual to focus attention on elementary row operations and their associated **elementary row matrices**.

Hence it is usual to use the term **elementary matrix** to mean exclusively an elementary row matrix.

When handling elementary column operations, the full term is then used: **elementary column matrix**.

On $\mathsf{Pr} \infty \mathsf{fWiki}$, where both treatments are discussed, it will be usual to use the full description for both types of **elementary matrix**.

## Also see

- Definition:Elementary Matrix Operation
- Definition:Elementary Row Operation
- Definition:Elementary Column Operation

- Results about
**elementary matrices**can be found**here**.

## Sources

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**elementary matrix**(abbrev.**E-matrix**) - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**elementary matrix**