  Let $F_n$ be an $n$ by $n$ symmetric matrix whose entries are bounded by
$n^{\gamma}$ for some $\gamma>0$. Consider a randomly perturbed matrix
$M_n=F_n+X_n$, where $X_n$ is a random symmetric matrix whose upper diagonal
entries $x_{ij}$ are iid copies of a random variable $\xi$. Under a very
general assumption on $\xi$, we show that for any $B>0$ there exists $A>0$ such
that $P(\sigma_n(M_n)\le n^{-A})\le n^{-B}$. The proof uses an inverse-type
result concerning concentration of quadratic forms, which is of interest of its
own.
