  We improve the current best running time value to invert sparse matrices over
finite fields, lowering it to an expected $O\big(n^{2.2131}\big)$ time for the
current values of fast rectangular matrix multiplication. We achieve the same
running time for the computation of the rank and nullspace of a sparse matrix
over a finite field. This improvement relies on two key techniques. First, we
adopt the decomposition of an arbitrary matrix into block Krylov and Hankel
matrices from Eberly et al. (ISSAC 2007). Second, we show how to recover the
explicit inverse of a block Hankel matrix using low displacement rank
techniques for structured matrices and fast rectangular matrix multiplication
algorithms. We generalize our inversion method to block structured matrices
with other displacement operators and strengthen the best known upper bounds
for explicit inversion of block Toeplitz-like and block Hankel-like matrices,
as well as for explicit inversion of block Vandermonde-like matrices with
structured blocks. As a further application, we improve the complexity of
several algorithms in topological data analysis and in finite group theory.
