We define the notions of trace, determinant and, more generally, Berezinian of matrices over a (Z₂)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 Dieudonne determinant of quaternionic matrices, but in general our quaternionic determinant is different.