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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