The kernel of the matrix $\lbrack i\mskip-2mu j \pmod n\rbrack$ when $n$ is prime

Abstract

In this paper, we consider the $n\times n$ matrix whose $\left(i,j\right)$-th entry is $ij\left(modn\right)$ and compute its rank and a basis for its kernel (viewed as a matrix over the real numbers) when $n$ is prime. We also give a conjecture on the rank of this matrix when $n$ is not prime and give a set of vectors in its kernel, which is a basis if the conjecture is true. Finally, we include an application of this problem to number theory.