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

PutAfter 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 $-a$. Thuswhere the coordinate vectors in the two blocks are understood. Expanding the determinant according to whether neither or both cross-block entries are selected givesThe 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.