  A cumbersome operation in many scientific fields, is inverting large
full-rank matrices. In this paper, we propose a coded computing approach for
recovering matrix inverse approximations. We first present an approximate
matrix inversion algorithm which does not require a matrix factorization, but
uses a black-box least squares optimization solver as a subroutine, to give an
estimate of the inverse of a real full-rank matrix. We then present a
distributed framework for which our algorithm can be implemented, and show how
we can leverage sparsest-balanced MDS generator matrices to devise matrix
inversion coded computing schemes. We focus on balanced Reed-Solomon codes,
which are optimal in terms of computational load; and communication from the
workers to the master server. We also discuss how our algorithms can be used to
compute the pseudoinverse of a full-rank matrix, and how the communication is
secured from eavesdroppers.
