Linear Algebra (2)

In a matrix a leading entry is the leftmost nonzero entry in a row. Consider the following matrix.

1   3   4   5   6
0   1   3   5   7
0   0   1   2   3

Rows 1, 2, and 3 have leading entries at columns 1, 2, and 3 respectively. A matrix is in row echelon form if all nonzero rows are above any rows of all zeros, each leading entry of a row is in a column to the right of the leading entry of the row above it, and all entries in a column below a leading entry are zeros. 

If a matrix is in row echelon form and each nonzero row has a leading entry of 1 and each leading 1 is the only nonzero entry in its column, then the matrix is said to be in reduced row echelon form. For example,

1   0   0   5   6
0   1   0   5   7
0   0   1   2   3

A pivot position in a matrix is a location that corresponds to a leading 1 in the reduced echelon form of the matrix. A pivot column is a column that contains a pivot position.

The Row Reduction Algorithm

Step 1
Begin with the leftmost nonzero. This is a pivot column. The pivot position is at the top.

Step 2
Select a nonzero entry in the pivot column as a pivot. If necessary, interchange rows to move this entry into the pivot position.

Step 3
Use row replacement operations to create zeros in all positions below the pivot.

Step 4
Repeat steps 1 - 3 to the submatrix that remains until there are no more nonzero rows to modify.

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