I solved the first question:
sage: me = m.extended_echelon_form()
sage: p = me.submatrix(0,3)
sage: p * m
[1 0 4]
[0 1 2]
[0 0 0]
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Hi,
It surprised me to find that my expectation fails:
sage: m = matrix(GF(5), 3, [1,2,3,4,5,6,7,8,9])
sage: m
[1 2 3]
[4 0 1]
[2 3 4]
sage: m.echelon_form(transformation=True)
[1 0 4]
[0 1 2]
[0 0 0]
sage: m = matrix(QQ, 3, [1,2,3,4,5,6,7,8,9])
sage: m.echelon_form(transformation=True)
[ 1 0
