Section 16.3 Essential Ideas

There are four elementary row operations for producing equivalent matrices:
1. RowSwap Interchange any two rows.
2. Row+ Row addition; add a row to any other row.
3.*RowScalar multiplication; multiply (or divide) all the elements of a row by the same nonzero real number.
4.*Row+Multiply all the entries of a row (pivot row) by a nonzero real number and add each resulting product to the corresponding entry of another specified row (target row).

These elementary row operations are used together in a process called pivoting: which means
1. Divide all entries in the row in which the pivot appears (called the pivot row) by the nonzero pivot element so that the pivot entry becomes a 1. This uses elementary row operation 3.
2. Obtain zeros above and below the pivot element by using elementary row operation 4.

GAUSS-JORDAN ELIMINATION

Step 1: Select as the first pivot the element in the first row, first column, and pivot.
Step 2: The next pivot is the element in the second row, second column; pivot.
Step 3: Repeat the process until you arrive at the last row, or until the pivot element is a zero. If it is a zero and you can interchange that row with a row below it, so that the pivot element is no longer a zero, do so and continue. If it is zero and you cannot interchange rows so that it is not a zero, continue with the next row. The final matrix is called the row-reduced form.