The sharp singularity rate for biased symmetric sign matrices

Let A${}_n$ be a symmetric random matrix whose entries on and above the diagonal are independent signs, equal to 1 with fixed probability $p\in (0,1)\setminus \{1/2\}$. We prove that $\mathbb{P}(\det A_n=0)=(p^2+(1-p{)}^2+o(1){)}^n$. The rate is attained by the event that two rows agree.