pivot position in rref

pivot position

In matrix A, the pivot position is the position corresponding to leading 1 in the row echelon form matrix (rref) of the matrix.
A pivot column is a column that contains a pivot position.

The rref of matrix A is ?
>>> from sympy import *
>>> A=Matrix([[0, -3, -6,4, 9],[-1,-2,-1, 3, 1], [-2, -3, 0, 3, -1], [1, 4, 5, -9, -7]])
>>> A
Matrix([
[ 0, -3, -6,  4,  9],
[-1, -2, -1,  3,  1],
[-2, -3,  0,  3, -1],
[ 1,  4,  5, -9, -7]])
>>> A.rref()
(Matrix([
[1, 0, -3, 0,  5],
[0, 1,  2, 0, -3],
[0, 0,  0, 1,  0],
[0, 0,  0, 0,  0]]), (0, 1, 3))

In that rref, the columns containing the leading 1 are 1, 2, and 4 columns.
$$\left[\begin{array}{rrrrr}1(pivot)&0&-3&0&5\\0&1(pivot)&2&0&-3\\0&0&0&1(pivot)&0 \\0&0&0&0&0\end{array}\right]$$


>>> from sympy import *
>>> B=Matrix([[0,3,-6,6,4,-5],[3,-7,8,-5,8,9],[3,-9,12,-9,6,15]])
>>> B
Matrix([
[0,  3, -6,  6, 4, -5],
[3, -7,  8, -5, 8,  9],
[3, -9, 12, -9, 6, 15]])
>>> B.rref()
(Matrix([
[1, 0, -2, 3, 0, -24],
[0, 1, -2, 2, 0,  -7],
[0, 0,  0, 0, 1,   4]]), (0, 1, 4))

In the case of matrix B
pivot position : [1,1], [2,2],[3,5]
pivot column : column 1, 2, 5