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Huh, that's weird. I tried it in several different ways and this is consistent.
I can tell you where the bottleneck is. It's in _echelon_in_place_classical. I called your matrices M and N, and:
sage: A1 = M.augment(M.parent().identity_matrix())
sage: time A1 = M.augment(M.parent().identity_matrix())
Time: CPU 0.00 s, Wall: 0.00 s
sage: time A2 = N.augment(N.parent().identity_matrix())
Time: CPU 0.00 s, Wall: 0.00 s
sage: time A1._echelon_in_place_classical()
Time: CPU 0.00 s, Wall: 0.01 s
sage: time A2._echelon_in_place_classical()
Time: CPU 0.24 s, Wall: 0.24 s
My (uninformed) guess is that there is a lot more Python that needs to be used with a "function" than with a "variable", slowing down the Cython. Anyone else?
