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. *Row
(Scalar 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.
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.