What a cool fact. The first thing that jumps into my mind for solving this would be the Schur Complement [1], which gives a way of computing the determinant of

What a cool fact. The first thing that jumps into my mind for solving this would be the Schur Complement [1], which gives a way of computing the determinant of a block matrix in terms of its components. This would give us a formula where we could express the determinant of block matrix as det([[A, B], [C, D]]) = det(A) det(D - CA^{-1}B) If we let A be the matrix of size n-1 by n-1, it seems like this would reduce to a statement about a sum of products of binomial coefficients equaling some other binomial coefficient. Not sure if there's a deeper reason though. [1]: https://en.wikipedia.org/wiki/Schur_complement#Properties https://en.wikipedia.org/wiki/Schur_complement#Properties