42 CHAPTER 2. SYSTEMS OF LINEAR EQUATIONS

Example 2.2.2 Here are some matrices which are in row reduced echelon form.

1 0 0 5 8 00 0 1 2 7 00 0 0 0 0 10 0 0 0 0 0

 ,

1 0 0 00 1 0 00 0 1 00 0 0 10 0 0 0

 .

Example 2.2.3 Here are matrices in echelon form which are not in row reduced echelonform but which are in echelon form.

1 0 6 5 8 20 0 2 2 7 30 0 0 0 0 10 0 0 0 0 0

 ,

1 3 5 40 2 0 70 0 3 00 0 0 10 0 0 0

Example 2.2.4 Here are some matrices which are not in echelon form.

0 0 0 01 2 3 30 1 0 20 0 0 10 0 0 0

 ,

 1 2 32 4 −64 0 7

 ,

0 2 3 31 5 0 27 5 0 10 0 1 0

 .

The following is the algorithm for obtaining a matrix which is in row reduced echelonform.

Algorithm 2.2.5

This algorithm tells how to start with a matrix and do row operations on it in such away as to end up with a matrix in row reduced echelon form.

1. Find the first nonzero column from the left. This is the first pivot column. Theposition at the top of the first pivot column is the first pivot position. Switch rows ifnecessary to place a nonzero number in the first pivot position.

2. Use row operations to zero out the entries below the first pivot position.

3. Ignore the row containing the most recent pivot position identified and the rows aboveit. Repeat steps 1 and 2 to the remaining sub-matrix, the rectangular array of numbersobtained from the original matrix by deleting the rows you just ignored. Repeat theprocess until there are no more rows to modify. The matrix will then be in echelonform.

4. Moving from right to left, use the nonzero elements in the pivot positions to zero outthe elements in the pivot columns which are above the pivots.