I want to find the maximal possible determinant of a $ m\times m$ - matrix A with entries $1..n$.
Conjecture 1 : The maximum possible determinant can be achieved by a matrix only conatining 1 and n.
Conjecture 2 : If A contains only 1 and n, then it can be writte in the form $$A~=~(n-1)H~+~\pmatrix{1..1\\.\ \ \ .\\.\ \ \ .\\1..1}$$ where H is a binary matrix. Sylvester's theorem gives $$det(A) = (n-1)^{m-1}(det(H)(n-1)+S)$$ where S is the sum of the entries of $det(H)\ H^{-1}$. For $m \le 20$, the maximum determinant of H is known, but unfortunately det(A) also depends on S. My conjecture is that the maximum possible S can be reached by a matrix H with maximum determinant.
- Can these conjectures be proved ?
- What is the largest possible value for S, given H has maximum determinant D ?