  We compute the rational Borel-Moore homology groups for affine determinantal
varieties in the spaces of general, symmetric, and skew-symmetric matrices,
solving a problem suggested by the work of Pragacz and Ratajski. The main
ingredient is the relation with Hartshorne's algebraic de Rham homology theory,
and the calculation of the singular cohomology of matrix orbits, using the
methods of Cartan and Borel. We also establish the degeneration of the
\v{C}ech-de Rham spectral sequence for determinantal varieties, and compute
explicitly the dimensions of de Rham cohomology groups of local cohomology with
determinantal support, which are analogues of Lyubeznik numbers first
introduced by Switala. Additionally, in the case of general matrices we further
determine the Hodge numbers of the singular cohomology of matrix orbits and of
the Borel-Moore homology of their closures, based on Saito's theory of mixed
Hodge modules.
