Inversion of two cyclotomic matrices

Abstract

Let $n\ge 3$ be a square-free natural number. We explicitly describe the inverses of the matrices $$ (2\sin(2\pi jk^*/n))_{j,k} \enspace \mbox{ and }\enspace (2\cos(2\pi jk^*/n))_{j,k}, $$ where $k^*$ denotes a multiplicative inverse of $k$ mod $n$ and $j,k$ run through the set $\{l; 1\le l\le n/2, (l,n)=1\}$. These results are based on the theory of Gauss sums.