Note on Hadamard's Determinant Theorem

Introduction. We shall call a square matrix A of order n an Hadamard matrix or for brevity an iî-matrix, if each element of A has the value ± 1 and if the determinant of A has the maximum possible value w. It is known that such a matrix A is an iï-matrix [ l] if, and only if, AA'~nEn where A f is the transpose of A and En is the unit matrix of order n. It is also known that, if an iï-matrix of order n > 1 exists, n must have the value 2 or be divisible by 4. The existence of an iî-matrix of order n has been proved [2,3] only for the following values of n>\\ (a) w = 2, (b) w = £*+ls~0 mod 4, p a prime, (c) n — m(p-\-l) where m ^ 2 is the order of an ü-matrix and p is a prime, (d) n = q(q — l) where q is a product of factors of types (a) and (b), (e) n = 172 and for n a product of any number of factors of types (a), (b), (c), (d) and (e). In this note we shall show that an iï-matrix of order n also exists when (f) n — q(q+3) where q and g+4 are both products of factors of types (a) and (b), (g) n = nin2(p+l)p, where Wi>l and w2>l are orders of i7-matrices and p is an odd prime, and (h) n — nin2in(tn+3) where Wi>l and W2>1 are orders of jff-matrices and m and ra+4 are both of the form p + l, p an odd prime. It is interesting to note the presence of the factors tii and w2 in the types (g) and (h) and their absence in the types (d) and (f).Thus, if p is a prime and £*+ls=0 mod 4, an iJ-matrix of order p(p+l) exists but, if p + l =2 mod 4, we can only be sure of the existence of an iJ-matrix of order nitt2p(p+l) where tii>l and ti2>l are orders of iï-matrices. This is analogous to the simpler result that, iî p + 1^0 mod 4 an ü-matrix of order p + l exists but, if p+lz=2 mod 4, we can only be sure of the existence of an i?-matrix of order n(p+l) where n > 1 is the order of an iï-matrix. We shall denote the direct product of two matrices A and B by A B and the unit matrix of order n by Ew.