This blog entry is to remind or introduce to people the fascinating problem called the Hadamard maximal determinant problem: What is the maximal possible determinant of a matrix M whose entries are of absolute value at most 1? Hadamard proved that if M is a complex matrix of order n, whose entries are bounded by $|M_{ij}| \leq 1$, for each i, j between 1 and n, then
$|det(M)| \leq n^{n/2}$ (equality is attained, so this is best possible for such matrices).

If instead the entries of the matrix are +1 or -1 and the size of the matrix is nxn where n is a multiple of 4, then the problem of the maximal determinant presumably boils down to the well-known search for Hadamard matrices. This is discussed in many books, papers and website but in particular, I refer to