# Definition:Gaussian Elimination

## Definition

Let $\mathbf A$ be a matrix over a field $K$.

Let $\mathbf E$ be a reduced echelon matrix which is row equivalent to $\mathbf A$.

The Gaussian elimination method is a technique for converting $\mathbf A$ into $\mathbf E$ by means of a sequence of elementary row operations.

## Also known as

Some sources refer to the technique as the Gauss elimination method.

## Also defined as

Some sources do not insist that $\mathbf E$ be a reduced echelon matrix at the end of the Gaussian elimination process, but merely a echelon matrix.

## Examples

### Arbitrary Matrix $1$

Let $\mathbf A$ denote the matrix:

$\mathbf A = \begin {bmatrix} 0 & 0 & 5 & 35 & -24 & 1 \\ 0 & 2 & 1 & -1 & 1 & 0 \\ 0 & 3 & 2 & 2 & -1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 5 & 3 & 1 & 0 & 1 \end {bmatrix}$

The reduced echelon form of $\mathbf A$ is:

$\mathbf E = \begin {bmatrix} 0 & 1 & 0 & -4 & 0 & 26 \\ 0 & 0 & 1 & 7 & 0 & -43 \\ 0 & 0 & 0 & 0 & 1 & -9 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end {bmatrix}$

### Arbitrary Matrix $2$

Let $\mathbf A$ denote the matrix:

$\mathbf A = \begin {bmatrix} 1 & -1 & 2 & 1 \\ 2 & 1 & -1 & 1 \\ 1 & -2 & 1 & 1 \\ \end {bmatrix}$

The reduced echelon form of $\mathbf A$ is:

$\mathbf E = \begin {bmatrix} 1 & 0 & 0 & \dfrac 5 8 \\ 0 & 1 & 0 & -\dfrac 1 8 \\ 0 & 0 & 1 & \dfrac 1 8 \\ \end {bmatrix}$

### Arbitrary Matrix $3$

Let $\mathbf A$ denote the matrix:

$\mathbf A = \begin {bmatrix} 1 & 1 & -1 \\ 1 & -1 & 2 \\ 2 & 0 & 2 \\ 2 & 1 & -1 \\ \end {bmatrix}$

The reduced echelon form of $\mathbf A$ is:

$\mathbf E = \begin {bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ \end {bmatrix}$

## Source of Name

This entry was named for Carl Friedrich Gauss.