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I'm not so sure that xy-xy can ever be non-zero and still provide arithmetic in which linear algebra operations make sense. For non-commuting variables, one can get at least something:
sage: A.<x,y>=FreeAlgebra(QQ)
sage: M=matrix([[x,y],[y*x,y^2]])
sage: M
[ x y]
[y*x y^2]
sage: M.determinant()
x*y^2 - y*x*y
