= Solution
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.
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