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 (_Z2)ⁿ${\left({\mathbb{Z}}_2\right)}^n$-graded commutative associative algebra A$A$. 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 (_Z2)ⁿ${\left({\mathbb{Z}}_2\right)}^n$-graded matrices of degree 0$0$ is polynomial in its entries. In the case of the algebra A=H$A=\mathbb{H}$ 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 (_Z2)ⁿ${\left({\mathbb{Z}}_2\right)}^n$-graded version of Liouville’s formula.