Monthly Archives: October 2015

Solution: 2015-18 Determinant

What is the determinant of the \(n\times n\) matrix \(A_n=(a_{ij}) \) where \[ a_{ij}=\begin{cases} 1 ,&\text{if } i=j, \\ x, &\text{if }|i-j|=1, \\ 0, &\text{otherwise,}\end{cases}\] for a real number \(x\)?

The best solution was submitted by Shin, Joonhyung (신준형, 2015학번). Congratulations!

Here is his soluton of problem 2015-18.

Alternative solutions were submitted by 김동률 (2015학번, +3), 박지민 (전산학부 석사 2015학번, +3), 최인혁 (2015학번, +3), 박성혁 (수리과학과 2014학번, +2), 박훈민 (수리과학과 2013학번, +2), 이상민 (수리과학과 2014학번, +2), 이영민 (수리과학과 2012학번, +2), 이종원 (수리과학과 2014학번, +2), 이호일 (수리과학과 2013학번, +2), 장기정 (수리과학과 2014학번, +2), 함도규 (2015학번, +2).

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

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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Solution: 2015-16 Complex integral

Evaluate the following integral for \( z \in \mathbb{C}^+ \).\[\frac{1}{2\pi} \int_{-2}^2 \log (z-x) \sqrt{4-x^2} dx.\]

The best solution was submitted by Lee, Jongwon (이종원, 수리과학과 2014학번). Congratulations!

Here is his solution of problem 2015-16.

Alternative solutions were submitted by 최인혁 (2015학번, +2), 박훈민 (수리과학과 2013학번, +2), 박성혁/이경훈 (수리과학과 2014학번, +2).

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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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