  We generalize linear superalgebra to higher gradings and commutation factors,
given by arbitrary abelian groups and bicharacters. Our central tool is an
extension, to monoidal categories of modules, of the Nekludova-Scheunert
faithful functor between the categories of graded-commutative and
supercommutative algebras. As a result we generalize (super-)trace, determinant
and Berezinian to graded matrices over graded-commutative algebras. For
instance, on homogeneous quaternionic matrices, we obtain a lift of the
Dieudonn\'e determinant to the skew-field of quaternions.
