Row Reduction (3)
From a General Matrix to an Echelon Form

• Start with the leftmost nonzero column of the matrix. We have nothing to do with the zero columns on the left of the working column; they are done.

• Pick a nonzero entry in the working column as a pivot

• Swap Row 1 and Row 3 to bring the pivot to the top of the working column

• Eliminating the entries below the pivot in the working column:
  Row 4 replaced by Row 4 - 3 Row 1
  Row 5 replaced by Row 5 - 2 Row 1

• The blue rows and columns are done. Now we focus on the remaining part, that is, the (Yellow) submatrix at the lower right corner.

• The next working column is the leftmost nonzero column within the (Yellow) submatrix. A nonzero entry in that column of the submatrix is picked as a new pivot. (If necessary, move this pivot to the top row of the submatrix by row swapping.)

• Eliminate the other entries in the (Yellow) working column by using the pivot:
  Row 3 replaced by Row 3 + (1/3)Row 2
  Row 4 replaced by Row 4 - (8/3)Row 2
  Row 5 replaced by Row 5 - (5/3)Row 2
  

• The blue part of the matrix is in good shape now. Our next working area is the Yellow submatrix at the lower right corner:

• The next working column is the leftmost nonzero column of the submatrix. The columns on the left of that column are all done.

• In the working column of the submatrix, pick a nonzero entry as the next working pivot. Move the wokring pivot to the top of the working column, by swapping Row 3 and Row 5.

• Using the working pivot to eliminate the other nonzero entries in the working column:
  Row 6 replaced by Row 6 + 2Row 3

• The resulting matrix is in echelon form now. The five pink entries are the pivots.

Remark 1: Even if we start from the same matrix, different choices of row operations may give different echelon forms. For example,

2 4 1 6	→ R2 replaced by R2-(1/2)R1	2 4 0 4	(Echelon Form)
2 4 1 6	→ Swap R1 & R2	1 6 2 4	→ R2 replaced by R2-2R1	1 6 0 -8	(Also an Echelon Form)

Remark 2: The reduced echelon form, however, is unique. The reduced echelon form is independent of the row operations you choose.