In this paper there is given a complete classification of the representations of the group $(2)*(2)$ (where $*$ denotes the free product) and the finite dihedral groups over a field of characteristic 2. These results follow from the solution of the problem of classifying to within similarity pairs of matrices $A,B$ (over a field) for which $A^2=0$ , $B^2=0$ . Bibliography: 5 titles.