  In this paper we provide concrete constructions of idempotents to represent
typical singular matrices over a given ring as a product of idempotents and
apply these factorizations for proving our main results. We generalize works
due to Laffey (Products of idempotent matrices. Linear Multilinear A. 1983) and
Rao (Products of idempotent matrices. Linear Algebra Appl. 2009) to
noncommutative setting and fill in the gaps in the original proof of Rao's main
theorems. We also consider singular matrices over B\'ezout domains as to when
such a matrix is a product of idempotent matrices.
