Linear Algebra with Applications, 10th Edition

1.2 Row Echelon Form

In Section 1.1, we learned a method for reducing an $n\times n$ linear system to strict triangular form. However, this method will fail if, at any stage of the reduction process, all the possible choices for a pivot element in a given column are 0.

Example 1

Consider the system represented by the augmented matrix

1.3-6 Full Alternative Text

If row operation III is used to eliminate the nonzero entries in the last four rows of the first column, the resulting matrix will be

1.3-7 Full Alternative Text

At this stage, the reduction ...