Recent advances in the construction of Hadamard matrices have depended on the existence of Baumert-Hall arrays and four (1, -1) matrices A, B, C, D of order m which are of Williamson type, that is pairwise satisfy
(i) MNT = NMT and
(ii) AAT + BBT + CCT + DDT = 4mlm.
If (i) is replaced by (i')MN = NM we have Goethals-Seidel matrices. These matrices are very important to the determination of the Hadamard conjecture: that there exists an Hadamard matrix of order 4t for all natural numbers t. This paper shows how the Williamson type and Goethals-Seidel type Hadamard matrices may be combined by introducing F-matrices which are a generalization of both Williamson and Goethals-Seidel matrices. Several constructions for F-matrices are given showing they exist for the new orders 119, 171, 185, 217 and the new classes 1/4q(q + I), q = 3(mod 8) a prime power and 1/2p(p - 3), p - 4(mod 4) and p - 4 both prime powers (among others).