Random symmetric matrices are almost surely nonsingular

Let Qn denote a random symmetric (n×n)-matrix, whose upper-diagonal entries are independent and identically distributed (i.i.d.) Bernoulli random variables (which take values 0 and 1 with probability 1/2). We prove that Qn is nonsingular with probability 1-O(n-1/8+δ) for any fixed δ>0. The proof uses a quadratic version of Littlewood-Offord-type results concerning the concentration functions of random variables and can be extended for more general models of random matrices