Exponential Rank Bounds for Random Matrices

Fix $b\in (0,1)$, let $1\le k\le n$, and let $A=(A_{ij})$ be an $n\times n$ random matrix with independent real entries satisfying $$\underset{x\in \mathbb{R}}{\sup }\mathbb{P}\{A_{ij}=x\}\le b<1(1\le i,j\le n)\mathrm{.}$$ We show that there exists $c>0$ such that $$\mathbb{P}\{\mathrm{rank}A\le n-k\}\le \exp (-cnk),1\le k\le n\mathrm{.}$$