  We employ the inductive structure of determinantal varieties to calculate the
weight filtration on local cohomology modules with determinantal support. We
show that the weight of a simple composition factor is uniquely determined by
its support and cohomological degree. As a consequence, we obtain the
equivariant structure of the Hodge filtration on each local cohomology module,
and we provide a formula for its generation level. In the case of square
matrices, we express the Hodge filtration in terms of the Hodge ideals for the
determinant hypersurface. As an application, we describe a recipe for
calculating the mixed Hodge module structure on any iteration of local
cohomology functors with determinantal support.
