Let $\{ e^i \}_{i=1}^{N}$ be the natural basis of $\mathbb{R}^N$ and denote $e^{N+1}\equiv e^1$. Define a matrix $A$ by
$$A \equiv \sum_{k=1}^{N} \alpha_k e^k \otimes e^k + \sum_{k=1}^{N} \alpha_{N+k} [ e^k \otimes e^k - e^k \otimes e^{k+1} - e^{k+1} \otimes e^k + e^{k+1} \otimes e^{k+1} ],$$
with $\otimes$ the Kronecker/outer product and the $\alpha$ some arbitrary constants. We can see that A is symmetric and almost tridiagonal, with the exception of the term $\propto e^{N} \otimes e^1 + e^1 \otimes e^N$. Should it be tridiagonal, there are established ways of computing $\det A$, see eg. this earlier question. Is there a way to compute the determinant in a similar way in this slightly more general case, or would my best bet be to extract the non-tridiagonal term (written as an outer product) and use the matrix determinant lemma?
