The Row Reduction Algorithm

(Gaussian elimination)

The forward phase

1. Begin with the leftmost nonzero column. This is a pivot column. The pivot position is at the top.
2. Select a nonzero entry in the pivot column as a pivot. If necessary, interchange rows to move this entry into the pivot position.
3. Use row replacement operations to create zeros in all positions below the pivot.
4. Cover (or ignore) the row containing the pivot position and cover all rows, if any, above it. apply steps 1-3 to the submatrix that remains. Repeat the process until there are no more nonzero rows to modify.

The backward phase

5. Beginning with the rightmost pivot and working upward and to the left, create zeros above each pivot. If a pivot is not 1, make it 1 by a scaling operation.

Remark: The forward phase produces a row echelon form of the input matrix. From this echelon form the backward phase produces the reduced row echelon form of the input matrix.