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Let $M_n$ denote a random symmetric $n$ by $n$ matrix, whose upper diagonalentries are iid Bernoulli random variables (which take value -1 and 1 withprobability 1/2). Improving the earlier result by Costello, Tao and Vu, we showthat $M_n$ is non-singular with probability $1-O(n^{-C})$ for any positiveconstant $C$. The proof uses an inverse Littlewood-Offord result for quadraticforms, which is of interest of its own.

Inverse Littlewood-Offord problems and The Singularity of Random Symmetric Matrices