If we reduce A to reduced echelon form, the leading entries are in the pivot positions of A.

It turns out that we can locate the pivot positions by reducing A to echelon form only.  The entries in these positions are called pivots, while the columns they appear in are called pivot columns.

For example, the matrix

3 -7 8 -5 8 9
3 -9 12 -9 6 15
0 3 -6 6 4 5

is row equivalent to

3 -9 12 -9 0 -9
0 2 -4 4 0 -14
0 0 0 0 1 4

and so we know that the pivot positions are those containing red numbers.  Also, the first, second and fifth columns are pivot columns.