  This paper addresses various questions about pairs of similarity classes of
matrices which contain commuting elements. In the case of matrices over finite
fields, we show that the problem of determining such pairs reduces to a
question about nilpotent classes; this reduction makes use of class types in
the sense of Steinberg and Green. We investigate the set of scalars that arise
as determinants of elements of the centralizer algebra of a matrix, providing a
complete description of this set in terms of the class type of the matrix.
  Several results are established concerning the commuting of nilpotent
classes. Classes which are represented in the centralizer of every nilpotent
matrix are classified--this result holds over any field. Nilpotent classes are
parametrized by partitions; we find pairs of partitions whose corresponding
nilpotent classes commute over some finite fields, but not over others. We
conclude by classifying all pairs of classes, parametrized by two-part
partitions, that commute. Our results on nilpotent classes complement work of
Ko\v{s}ir and Oblak.
