Note on Hadamard’s determinant theorem

Abstract

Introduction. We shall call a square matrix A of order n an Hadamard matrix or for brevity an i-matrix, if each element of A has the value 1 and if the determinant of A has the maximum possible value w n/2 . It is known that such a matrix A is an i-matrix [l] 1 if, and only if, AA'~nE n where A f is the transpose of A and E n is the unit matrix of order n. It is also known that, if an i-matrix of order n > 1 exists, n must have the value 2 or be divisible by 4. The existence of an i-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 h -\-l) where m^2 is the order of an -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).