Definition:Gaussian Elimination

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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 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 an echelon matrix.


Also known as

Some sources refer to the technique of Gaussian elimination as Gauss elimination.

Gaussian elimination is sometimes seen referred to as pivotal condensation.


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:

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


Also see

  • Results about Gaussian elimination can be found here.


Source of Name

This entry was named for Carl Friedrich Gauss.


Sources