Norm of the inverse of a random matrix

Let A be an n times n matrix, whose entries are independent copies of a centered random variable satisfying the subGaussian tail estimate. We prove that the operator norm of´ A^-1 does not exceed Cn^3/2 with probability close to 1. In a geometric language, this bounds the probability that the affine span of n random vectors in Ropfⁿ with i.i.d. subGaussian coordinates comes close to the origin