  We extend probability estimates on the smallest singular value of random
matrices with independent entries to a class of sparse random matrices. We show
that one can relax a previously used condition of uniform boundedness of the
variances from below. This allows us to consider matrices with null entries or,
more generally, with entries having small variances. Our results do not assume
identical distribution of the entries of a random matrix and help to clarify
the role of the variances of the entries. We also show that it is enough to
require boundedness from above of the $r$-th moment, $r > 2$, of the
corresponding entries.
