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In this paper we use the results from the first part to compute the vanishingtopology for matrix singularities based on certain spaces of matrices. We placethe variety of singular matrices in a geometric configuration of free divisorswhich are the "exceptional orbit varieties" for repesentations of solvablegroups. Because there are towers of representations for towers of solvablegroups, the free divisors actually form a tower of free divisors $E_n$, and wegive an inductive procedure for computing the vanishing topology of the matrixsingularities. The inductive procedure we use is an extension of thatintroduced by L\^{e}-Greuel for computing the Milnor number of an ICIS. Insteadof linear subspaces, we use free divisors arising from the geometricconfiguration 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 homotopyequivalent to a bouquet of spheres, whose number is called the singular Milnornumber. We give formulas for this singular Milnor number in terms of singularMilnor numbers of various free divisors on smooth subspaces, which can becomputed as lengths of determinantal modules. In addition to being applied tosymmetric, general and skew-symmetric matrix singularities, the results arealso applied to Cohen--Macaulay singularities defined as 2 x 3 matrixsingularities. We compute the Milnor number of isolated Cohen--Macaulay surfacesingularities of this type in $\mathbb{C}^4$ and the difference of Bettinumbers of Milnor fibers for isolated Cohen--Macaulay 3--fold singularities ofthis type in $\mathbb{C}^5$.

Solvable Groups, Free Divisors and Nonisolated Matrix Singularities II: Vanishing Topology