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 \( i, j = 1, 2, \cdots, N-1 \).

The best solution was submitted by Park, Hun Min (박훈민, 수리과학과 2013학번). Congratulations!

Here is his solution of problem_2015_17.

Alternative solutions were submitted by 이종원 (수리과학과 2014학번, +3), 최인혁 (2015학번, +3), 박성혁 (수리과학과 2014학번, +3, solution), 신준형 (2015학번, +3), 이영민 (수리과학과 2012학번, +3), 장기정 (수리과학과 2014학번, +3), 함도규 (2015학번, +3).

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