  We consider a class of sparse random matrices of the form $A_n
=(\xi_{i,j}\delta_{i,j})_{i,j=1}^n$, where $\{\xi_{i,j}\}$ are i.i.d.~centered
random variables, and $\{\delta_{i,j}\}$ are i.i.d.~Bernoulli random variables
taking value $1$ with probability $p_n$, and prove a quantitative estimate on
the smallest singular value for $p_n = \Omega(\frac{\log n}{n})$, under a
suitable assumption on the spectral norm of the matrices. This establishes the
invertibility of a large class of sparse matrices. For $p_n =\Omega(
n^{-\alpha})$ with some $\alpha \in (0,1)$, we deduce that the condition number
of $A_n$ is of order $n$ with probability tending to one under the optimal
moment assumption on $\{\xi_{i,j}\}$. This in particular, extends a conjecture
of von Neumann about the condition number to sparse random matrices with
heavy-tailed entries. In the case that the random variables $\{\xi_{i,j}\}$ are
i.i.d.~sub-Gaussian, we further show that a sparse random matrix is singular
with probability at most $\exp(-c n p_n)$ whenever $p_n$ is above the critical
threshold $p_n = \Omega(\frac{ \log n}{n})$. The results also extend to the
case when $\{\xi_{i,j}\}$ have a non-zero mean. We further find quantitative
estimates on the smallest singular value of the adjacency matrix of a directed
Erd\H{o}s-R\'{e}yni graph whenever its edge connectivity probability is above
the critical threshold $\Omega(\frac{\log n}{n})$.
