PIVOT POSITION

§If a matrix A is row equivalent to an echelon matrix U, we call U an echelon form (or row echelon form) of A; if U is in reduced echelon form, we call U the reduced echelon form of A.

§

§A pivot position in a matrix A is a location in A that corresponds to a leading 1 in the reduced echelon form of A. A pivot column is a column of A that contains a pivot position.

Example 1: Row reduce the matrix A below to echelon form, and locate the pivot columns of A.

Solution: The top of the leftmost nonzero column is the first pivot position. A nonzero entry, or pivot, must be placed in this position.

§Now, interchange rows 1 and 4.

§Create zeros below the pivot, 1

, by adding multiples of the first 

row to the rows below, 

and obtain 

the next matrix. 

§Choose 2 in the second row

 as the next pivot.

§Add  -5/2   times row 2 to row 3, and add 3/2 times row 2 to row 4.

§There is no way a leading entry can be created in column 3. But, if we interchange rows 3 and 4, we can produce a leading entry in column 4.

§The matrix is in echelon form and thus reveals that columns 1, 2, and 4 of A are pivot columns.

§The pivots in the example are 1, 2 and     .