Random symmetric matrices are almost surely nonsingular

Let Q n 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 Q n 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