Summary: For any natural number \(n\) and \((m,n)=1\), we analyse the eigenvalues and their multiplicities of the matrix \(\mathcal{A}(n,m) := (\zeta_n^{mrs})\) for \(0\le r,s \le n-1\). As a consequence, we evaluate the quadratic Gauss sum and derive the law of quadratic reciprocity using only elementary methods.