Let $1\le r<n$, and let $J$ denote the set of $r$-tuples $\mathbf{j}=(j_1,\ldots,j_r)$ with $1\le j_1,\ldots,j_r\le n$ and $j_s\ne j_t$ for all $s\ne t$. Consider the $P_n^r\times P_n^r$ matrix $A$ (the rows and columns being indexed by $J$) whose $(\mathbf{j},\mathbf{k})$-entry is $\zeta_n^{\mathbf{j}\cdot\mathbf{k}}$, where $\zeta_n=\exp(2\pi i/n)$, and $$\mathbf{j}\cdot\mathbf{k}=j_1k_1+\cdots+j_rk_r.$$
I want to know whether there is a beautiful formula for $\det(A)$, or at least whether $A$ is invertible. If $A$ is invertible, is there an explicit formula for its inverse?
Thanks in advance!