Inversion of two cyclotomic matrices

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