---
tag: 098D6FQN3N
kind: concept
name: Gauss–Jordan elimination
msc: [15A06, 65F05]
sources:
  - url: https://en.wikipedia.org/wiki/Gaussian_elimination
    locator: lead section
---

## Summary

Gauss–Jordan elimination is the variant of [[Gaussian elimination|0JVJ2WCXZH]] that continues until the matrix is in [[reduced row echelon form|0WF5AQYTW6]].
After the forward phase gives a row echelon form, it scales each pivot row to make the [[pivot|0J0BBCMJ5W]] equal to $1$.
Then it adds multiples of each pivot row to the rows above it, to make every other entry in each pivot column zero.
Applied to the [[augmented matrix|0T6MWGZSXA]] of a system, the result shows the solutions directly, with no [[back substitution|0ATFVE08MW]].

## Remarks

The result of Gauss–Jordan elimination does not depend on the choices in the algorithm.
The reason is that the reduced row echelon form of a matrix is unique.
Applied to $[A \mid I]$ for an invertible matrix $A$, it gives $[I \mid A^{-1}]$, see [[computing the inverse by row reduction|0Z83DHFMYM]].

## Explanations

### L1 human

Gauss–Jordan elimination cleans up the staircase matrix that Gaussian elimination produces.
It makes every pivot equal to $1$ and clears the entries above the pivots too.
Then each equation names one basic unknown in terms of the free unknowns.