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Determinantshard
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For a 4×44 \times 44×4 Schur complement problem: Let M=(ABCD)M = \begin{pmatrix} A & B \\ C & D \end{pmatrix}M=(AC​BD​) where AAA is 2×22 \times 22×2, BBB is 2×22 \times 22×2, CCC is 2×22 \times 22×2, DDD is 2×22 \times 22×2, and AAA is invertible. The Schur complement is S=D−CA−1BS = D - CA^{-1}BS=D−CA−1B, with det⁡(M)=det⁡(A)det⁡(S)\det(M) = \det(A)\det(S)det(M)=det(A)det(S). If det⁡(A)=2\det(A) = 2det(A)=2, det⁡(B)=3\det(B) = 3det(B)=3, det⁡(C)=−1\det(C) = -1det(C)=−1, and det⁡(S)=5\det(S) = 5det(S)=5, what is det⁡(M)\det(M)det(M)?