# 2015-17 Inverse of a minor

Let $$H$$ be an $$N \times N$$ positive definite matrix and $$G = H^{-1}$$. Let $$H’$$ be an $$(N-1) \times (N-1)$$ matrix obtained by removing the $$N$$-th row and the column of $$H$$, i.e., $$H’_{ij} = H_{ij}$$ for any $$i, j = 1, 2, \cdots, N-1$$. Let $$G’ = (H’)^{-1}$$. Prove that
$G_{ij} – G’_{ij} = \frac{G_{iN} G_{Nj}}{G_{NN}}$
for any $$i, j = 1, 2, \cdots, N-1$$.

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