Let GF(q) denote a finite field of order q = py, p a prime. Let A and C be symmetric matrices of order n, rank m and order s, rank k, respectively, over GF(q). Carlitz  has determined the number N(A, C, n, s) of solutions X over GF(q), for p an odd prime, to the matrix equation
where n = m. Furthermore, Hodges  has determined the number N(A, C, n, s, r) of s × n matrices X of rank r over GF(q), p an odd prime, which satisfy (1.1). Perkin  has enumerated the s × n matrices of given rank r over GF(q), q = 2y, such that XXT = 0. Finally, the author  has determined the number of solutions to (1.1) in case C = 0, where q = 2y.