On the superdeterminant function for supermatrices
Nigel B. Backhouse, Anargyros Fellouris · Journal of Physics A Mathematical and General · 1984
A rigorous proof is given of the multiplicative property of the superdeterminant (Berezinian) of a supermatrix. The proof obviates the need to consider the domain of existence of the logarithm function for supermatrices, and devolves on the identity det(1-PQ) det(1-QP)=1, where P and Q are compatible rectangular matrices over the odd part of a Grassmann algebra.