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You can use the echelon_form method with transformation=True.
Example:
sage: A = Matrix([[1, 2, 3], [3, 2, 1], [1, 1, 1]])
sage: A
[1 2 3]
[3 2 1]
[1 1 1]
sage: A.echelon_form(transformation=True)
(
[ 1 0 -1] [ 0 1 -2]
[ 0 1 2] [ 0 -1 3]
[ 0 0 0], [ 1 1 -4]
)
