  We develop the theory of linear algebra over a (Z_2)^n-commutative algebra (n
in N), which includes the well-known super linear algebra as a special case
(n=1). Examples of such graded-commutative algebras are the Clifford algebras,
in particular the quaternion algebra H. Following a cohomological approach, we
introduce analogues of the notions of trace and determinant. Our construction
reduces in the classical commutative case to the coordinate-free description of
the determinant by means of the action of invertible matrices on the top
exterior power, and in the supercommutative case it coincides with the
well-known cohomological interpretation of the Berezinian.
