Given $n\ge 2\ \ ,\ u,v,k\ $ integers.
Decision problem :
Does a $n\times n$ - matrix with entries from $u$ to $v$ with determinant $k$ exist?
In which complexity class is this problem ? Is it NP-hard?
A similar problem
Given $n\ge 2\ ,\ k\ $ integer
Is there a latin square matrix of size $n \times n$ with determinant $k$?
What about this problem ?
It is clear that the problems are in NP because the determinant of a given matrix can be computed in polynomial time. But I have no idea how to check how hard it is to find a matrix with the desired properties or proof that there is none.