A [b]pivot position[/b] in a matrix [math]A[/math] is a location in [math]A[/math] that corresponds to a leading entry (a leading "1") when [math]A[/math] is row reduced to a matrix in echelon form (reduced echelon form).[br][br]A [b]pivot column[/b] is a column that contains a pivot position.[br][br]A [b]pivot[/b] is a nonzero number in a pivot position that is used as needed to create zeros in the column that contains that nonzero number through the use of replacement row operations.[br][br]We will introduce a systematic method called the [b]row reduction algorithm[/b] to locate all the pivot positions one by one and transform the matrix to the one in echelon form (and reduced echelon form).[br] [br]

The row reduction algorithm consists of four steps, and it produces a matrix in echelon form. The last step produces a matrix in reduced echelon form.[br][br][u]Step 1[/u]: Begin with the leftmost nonzero column. This is a pivot column. The pivot position is at the top.[br][br][u]Step 2[/u]: Select a nonzero entry in the pivot column as a pivot. If necessary, interchange rows to move this entry into the pivot position.[br][br][u]Step 3[/u]: Use row replacement operations to create zeros in all positions below the pivot.[br][br][u]Step 4[/u]: Ignore the row containing the pivot position and all rows, if any, above it. Apply steps 1-3 to the submatrix that remains. Repeat the process until there are no more nonzero rows to modify.[br][br][br][u]Example[/u]: Given the following augmented matrix[br][br][math]\left(\begin{array}{ccccc|c} 0 & 0 & 0 & 2 & 1 & 9 \\0 & -2 & -6 & 2 & 0 & 2 \\ [br]0 & 2 & 6 & -2 & 2 & 0 \\ 0 & 3 & 9 & 2 & 2 & 19\end{array}\right)[/math][br][br]We will use the "[url=http://www.math.odu.edu/~bogacki/cgi-bin/lat.cgi?c=roc]row operation calculator[/url]" again to demonstrate the row reduction algorithm.[br]