The analogous question for matrices with elements equal to 0 or 1 is equivalent since, as will be shown below, the maximal determinant of a {1,−1} matrix of size n is 2n−1 times the maximal determinant of a {0,1} matrix of size n−1.
The problem was posed by Hadamard in the 1893 paper[1] in which he presented his famous determinant bound and remains unsolved for matrices of general size.
Additional examples were later constructed by Scarpis and Paley and subsequently by many other authors.
The earliest results are due to Barba, who tightened Hadamard's bound for n odd, and Williamson, who found the largest determinants for n=3, 5, 6, and 7.
Some important results include The design of experiments in statistics makes use of {1, −1} matrices X (not necessarily square) for which the information matrix XTX has maximal determinant.
For a {1, −1} matrix, it means any two rows differ in exactly half of the entries, which is impossible when n is an odd number.
The determinants of equivalent matrices are equal, except possibly for a sign change, and it is often convenient to standardize R by means of negations and permutations of rows and columns.
Hadamard's bound can be derived by noting that |det R| = (det G)1/2 ≤ (det nI)1/2 = nn/2, which is a consequence of the observation that nI, where I is the n by n identity matrix, is the unique matrix of maximal determinant among matrices satisfying properties 1–4.
That det R must be an integer multiple of 2n−1 can be used to provide another demonstration that Hadamard's bound is not always attainable.
The dot product of two rows of the same type is congruent to n (mod 4); the dot product of two rows of opposite type is congruent to n+2 (mod 4).
Ehlich showed that if R attains the bound, and if the rows and columns of R are permuted so that both G = RRT and G′ = RTR have the standard form and are suitably normalized, then we may write where W, X, Y, and Z are (n/2)×(n/2) matrices with constant row and column sums w, x, y, and z that satisfy z = −w, y = x, and w2+x2 = 2n−2.
Ehlich[7] showed that when n ≡ 3 (mod 4), the strengthened property 1 implies that the maximal-determinant form of G can be written as B−J where J is the all-one matrix and B is a block-diagonal matrix whose diagonal blocks are of the form (n-3)I+4J.
This optimal form leads to the bound where v = n−rs is the number of blocks of size r+1 and u =s−v is the number of blocks of size r. Cohn[8] analyzed the bound and determined that, apart from n = 3, it is an integer only for n = 112t2±28t+7 for some positive integer t. Tamura[9] derived additional restrictions on the attainability of the bound using the Hasse–Minkowski theorem on the rational equivalence of quadratic forms, and showed that the smallest n > 3 for which Ehlich's bound is conceivably attainable is 511.