Let $n\ge 1$. Consider an $n\times n$ matrix where the $k$th row of the matrix has entries $k, k+1,\cdots, n, 1,2,\cdots, k-1$ in that order. Find the determinant of this matrix. For instance when $n=2$ the matrix is $\begin{pmatrix}1 & 2\\ 2 & 1\end{pmatrix}$
Let $R_i$ be the ith row and $C_i$ be the ith column. Performing the operation $C_i\mapsto C_i - C_{1}$ for $2\leq i\leq n$ yields a matrix whose first column consists of the entries $1,2,\cdots, n$, and the kth column has $-(n-1)$ as the $n-k+2$th entry and $1$'s elsewhere. Then perform the operations $R_i\mapsto R_i-R_1$ for $2\leq i\leq n$ to get a matrix where the first column equals $(1,1,2,\cdots, n-1)^T$ and the kth column starts with a 1, has a $-1$ in the $n-k+2$th entry and zeroes elsewhere.
I'm not sure how to proceed from here. It might be possible to factor out a common factor to compute the determinant.
Edit: here's a link to a relevant post about anti-circulant matrices: Eigenvalues of anti-circulant matrix.