On decomposing any matrix as a linear combination of three idempotents

In a recent article, we gave a full characterization of matrices that can be decomposed as linear combinations of two idempotents with prescribed coefficients. In this one, we use those results to improve on a recent theorem of Rabanovich: we establish that every square matrix is a linear combination of three idempotents (for an arbitrary coefficient field rather than just one of characteristic 0).