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:
- $\mathbf E = \begin {bmatrix}
1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ \end {bmatrix}$
Also see
Source of Name
This entry was named for Carl Friedrich Gauss.
Sources
- 1982: A.O. Morris: Linear Algebra: An Introduction (2nd ed.) ... (previous) ... (next): Chapter $1$: Linear Equations and Matrices: $1.2$ Elementary Row Operations on Matrices: Theorem $1.5$