  Let $H$ and $K$ be groups. In this paper we introduce a concept of
determinant for automorphisms of $H\times K$ and some concepts of
incompatibility for group pairs as a measure of how much $H$ and $K$ are fare
from being isomorphic. With the aid of the tools developed from these
definitions, we give a characterisation of invertible automorphisms of $H\times
K$ by means of their determinants and an explicit description of Aut($H\times
K$) as a group of $2$-by-$2$ matrices, in case $H$ or $K$ belong to some
relevant classes of groups. Many theoretical and practical applications of the
determinants will be presented, together with examples and an analysis on some
computational advantages of the determinants.
