NUMBER OF RANK r SYMMETRIC MATRICES OVER FINITE FIELDS

We determine the number of n×n symmetric matrices over GF (p) that have rank r for 0 ≤ r ≤ n. In [BM2] Brent and McKay determine the number of n × n symmetric matrices over Zp that have determinant zero. Thus they determine the number of n× n symmetric matrices over Zp that have rank n. We extend their result to symmetric matrices over GF (p) and we determine the number of matrices that have rank r for any r. The problem when the matrix is not required to be symmetric was treated in [BM1] and in [GR]. In these papers the number of (n+∆)×n matrices over Zp with rank r is determined for all r and ∆ ≥ 0. Let I(n, r, p) be the number of n × n symmetric matrices over GF (p) with rank r. Furthermore, let q(n, p) be the probability that an n× n symmetric matrix over GF (p) is invertible. Define q(0, p) to be 1. Also note that q(1, p) = (1− 1 pk ). Theorem 0.1. In the notation given above, q(n, p) satisfies the recurrence (0.1) q(n, p) = ( 1− ( 1 pk )) q(n− 1, p) + ( 1 pk ) ( 1− ( 1 pk )n−1) q(n− 2, p) for all n ≥ 2. Furthermore, this recurrence gives