Put
$$
a=2\cos\left(\frac{\pi}{m_{ij}}\right).
$$
After ordering $I_1$ before $I_2$, the only nonzero off-diagonal entries between the two blocks occur at $(i,j)$ and $(j,i)$, where they equal $-a$. Thus
$$
G(W)=
\begin{pmatrix}
G(W_1)&-a,e_ie_j^T\\
-a,e_je_i^T&G(W_2)
\end{pmatrix},
$$
where the coordinate vectors in the two blocks are understood. Expanding the <determinant> according to whether neither or both cross-block entries are selected gives
$$
\det G(W)
=\det G(W_1)\det G(W_2)
-a^2\det G(W_1')\det G(W_2').
$$
The minus sign is the sign of the transposition pairing the two cross-block entries. This formula remains valid when either diagonal block is singular, so no inverse or <Schur complement> is needed.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 111 / 2 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-111/2/c/solution