Gauss–Jordan elimination

Summary

Gauss–Jordan elimination is the variant of Gaussian elimination that continues until the matrix is in reduced row echelon form. After the forward phase gives a row echelon form, it scales each pivot row to make the pivot 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 of a system, the result shows the solutions directly, with no back substitution.

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.

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.