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We show that for an $n\times n$ random matrix $A$ with independent uniformly anti-concentrated entries, such that $\mathbb{E} ||A||^2_{HS}\leq K n^2$, the smallest singular value $\sigma_n(A)$ of $A$ satisfies $$ P\left( \sigma_n(A)\leq \frac{\varepsilon}{\sqrt{n}} \right) \leq C\varepsilon+2e^{-cn},\quad \varepsilon \ge 0.
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