A large deviation inequality for the rank of a random matrix (Q6618736)

The author considers estimating the probability that an \(n\times n\) random matrix with independent identically distributed (i.i.d.) entries is singular, which is a classical problem in probability. More specifically, for an \(n\times n\) random matrix \(A\) with independent identically distributed nonconstant sub-Gaussian entries it is proven that for any natural \(k \leq c\sqrt n\) holds \(\mathrm{rank}(A) \geq n-k\) with probability at least \(1-e^{(-c'kn)}\), where \(c\) and \(c'\) are constants.