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  "body": "\n## Summary\n\nGauss–Jordan elimination is the variant of [[Gaussian elimination|0JVJ2WCXZH]] that continues until the matrix is in [[reduced row echelon form|0WF5AQYTW6]].\nAfter the forward phase gives a row echelon form, it scales each pivot row to make the [[pivot|0J0BBCMJ5W]] equal to $1$.\nThen it adds multiples of each pivot row to the rows above it, to make every other entry in each pivot column zero.\nApplied to the [[augmented matrix|0T6MWGZSXA]] of a system, the result shows the solutions directly, with no [[back substitution|0ATFVE08MW]].\n\n## Remarks\n\nThe result of Gauss–Jordan elimination does not depend on the choices in the algorithm.\nThe reason is that the reduced row echelon form of a matrix is unique.\nApplied to $[A \\mid I]$ for an invertible matrix $A$, it gives $[I \\mid A^{-1}]$, see [[computing the inverse by row reduction|0Z83DHFMYM]].\n\n## Explanations\n\n### L1 human\n\nGauss–Jordan elimination cleans up the staircase matrix that Gaussian elimination produces.\nIt makes every pivot equal to $1$ and clears the entries above the pivots too.\nThen each equation names one basic unknown in terms of the free unknowns.\n",
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