  We study codimension two determinantal varieties with isolated singularities.
These singularities admit a unique smoothing, thus we can define their Milnor
number as the middle Betti number of their generic fiber. For surfaces in C^4,
we obtain a L\^e-Greuel formula expressing the Milnor number of the surface in
terms of the second polar multiplicity and the Milnor number of a generic
section. We also relate the Milnor number with Ebeling and Gusein-Zade index of
the 1- form given by the differential of a generic linear projection defined on
the surface. To illustrate the results, in the last section we compute the
Milnor number of some normal forms from A. Fr\"uhbis-Kr\"uger and A. Neumer [2]
list of simple determinantal surface singularities.
