Higher trace and Berezinian of matrices over a Clifford algebra

We define the notions of trace, determinant and, more generally, Berezinian of matrices
over a (Z_2)^n-graded commutative associative algebra A. The applications include a new
approach to the classical theory of matrices with coefficients in a Clifford algebra, in
particular of quaternionic matrices. In a special case, we recover the classical Dieudonné
determinant of quaternionic matrices, but in general our quaternionic determinant is
different. We show that the graded determinant of purely even (Z_2)^n-graded matrices of
degree 0 is polynomial in its entries. In the case of the algebra A = H of quaternions, we
calculate the formula for the Berezinian in terms of a product of quasiminors in the sense
of Gelfand, Retakh, and Wilson. The graded trace is related to the graded Berezinian (and
determinant) by a (Z_2)^n-graded version of Liouville’s formula.