  In this paper we use the results from the first part to compute the vanishing
topology for matrix singularities based on certain spaces of matrices. We place
the variety of singular matrices in a geometric configuration of free divisors
which are the "exceptional orbit varieties" for repesentations of solvable
groups. Because there are towers of representations for towers of solvable
groups, the free divisors actually form a tower of free divisors $E_n$, and we
give an inductive procedure for computing the vanishing topology of the matrix
singularities. The inductive procedure we use is an extension of that
introduced by L\^{e}-Greuel for computing the Milnor number of an ICIS. Instead
of linear subspaces, we use free divisors arising from the geometric
configuration and which correspond to subgroups of the solvable groups.
  Here the vanishing topology involves a singular version of the Milnor fiber;
however, it still has the good connectivity properties and is homotopy
equivalent to a bouquet of spheres, whose number is called the singular Milnor
number. We give formulas for this singular Milnor number in terms of singular
Milnor numbers of various free divisors on smooth subspaces, which can be
computed as lengths of determinantal modules. In addition to being applied to
symmetric, general and skew-symmetric matrix singularities, the results are
also applied to Cohen--Macaulay singularities defined as 2 x 3 matrix
singularities. We compute the Milnor number of isolated Cohen--Macaulay surface
singularities of this type in $\mathbb{C}^4$ and the difference of Betti
numbers of Milnor fibers for isolated Cohen--Macaulay 3--fold singularities of
this type in $\mathbb{C}^5$.
