A particular matrix with exponential form, its inversion and some norms

In this paper, we study a particular $n\times n$ matrix $A=[a_{k_{ij}}{]}_{i,j=1}^n$ and its Hadamard inverse $A^{\circ (-1)}$, whose entire elements are exponential form $a_k=e(\frac{k}{n})=e^{\frac{2\pi ik}{n}},$ where $k_{ij}=min(i,j)+1$. We study determinants, leading principal minor and inversions of $A,$ $A^{\circ (-1)}$. Then the defined values of Euclidean norms, $l_p$ norms and spectral norms of these matrices are presented, rather than upper and lower bounds, which are different from other articles.