The least singular value of a random square matrix is O(n^{-1/2})
  Let A be a matrix whose entries are real i.i.d. centered random variables with unit variance and suitable moment assumptions. Then the smallest singular value of A is of order n^{-1/2} with high probability. The lower estimate of this type was proved recently by the authors; in this note we establish the matching upper estimate. 
Mark Rudelson and Roman Vershynin