  Let $A$ be an $n\times n$ real matrix, and let $M$ be an $n\times n$ random
matrix whose entries are i.i.d sub-Gaussian random variables with mean $0$ and
variance $1$. We make two contributions to the study of $s_n(A+M)$, the
smallest singular value of $A+M$.
  (1) We show that for all $\epsilon \geq 0$,
  $$\mathbb{P}[s_n(A + M) \leq \epsilon] = O(\epsilon \sqrt{n}) +
2e^{-\Omega(n)},$$ provided only that $A$ has $\Omega (n)$ singular values
which are $O(\sqrt{n})$. This extends a well-known result of Rudelson and
Vershynin, which requires all singular values of $A$ to be $O(\sqrt{n})$.
  (2) We show that any bound of the form
  $$\sup_{\|{A}\|\leq n^{C_1}}\mathbb{P}[s_n(A+M)\leq n^{-C_3}] \leq n^{-C_2}$$
must have $C_3 = \Omega (C_1 \sqrt{C_2})$. This complements a result of Tao and
Vu, who proved such a bound with $C_3 = O(C_1C_2 + C_1 + 1)$, and counters
their speculation of possibly taking $C_3 = O(C_1 + C_2)$.
