Introduction. Let GF{q) denote a finite field of order q = p y , 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 [6] 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 [9] has determined the number N(A, C, n, s, r) of s X n matrices X of rank r over GF(q), p an odd prime, which satisfy (1.1). Perkin [10] has enumerated the s X n

more »

... es of given rank r over GF(q), q = 2 V , such that XX T = 0. Finally, the author [3] has determined the number of solutions to (1.1) in case C = 0, where q = 2 y . An n X n symmetric matrix over GF(2 V ) is said to be an alternate matrix if A has 0 diagonal. Otherwise, A is said to be nonalternate. The author [4; 5] has determined the number N(A, C, n, s, r) of s X n matrices X of rank r over GF(q), q = 2 y , which satisfy (1.1), in case A is an alternate matrix over GF(q) and in case both A and C are symmetric, nonalternate matrices over GF{q). The purpose of this paper is to determine the number N(A, C, n, s, r), in case A is a symmetric, nonalternate matrix over GF(2 V ) and C is an alternate matrix over GF(2 V ). In determining this number, Albert's canonical forms for symmetric matrices over fields of characteristic two are used [1]. These forms and other necessary preliminaries appear in Section 2. In Section 3, the number N(A, C, n, s) is found, in case both A and C are nonsingular. Finally, in Section 4, the number N(A, C, n, s, r), 0 ^ r ^ min (s, n), is determined. The difference equations obtained in Section 4 were solved by using methods due to Carlitz [7] . Throughout the remainder of this paper, GF(q) will denote a finite field of order q = 2 V and V n will denote an w-dimensional vector space over GF(q). Further, for any matrix M over GF(q) f 3% Sf [M] will denote the row space of M. For matrices X x , X 2 , . . . , X k , where X t is nii X n, col [X\, X 2 , . . . , X k ]