# 2012-24 Determinant of a Huge Matrix

Consider all non-empty subsets $$S_1,S_2,\ldots,S_{2^n-1}$$ of $$\{1,2,3,\ldots,n\}$$. Let $$A=(a_{ij})$$ be a $$(2^n-1)\times(2^n-1)$$ matrix such that $a_{ij}=\begin{cases}1 & \text{if }S_i\cap S_j\ne \emptyset,\\0&\text{otherwise.}\end{cases}$ What is $$\lvert\det A\rvert$$?

(This is the last problem of this semester. Good luck with your final exam!)

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# Solution: 2012-22 Simple integral

Compute $$\int_0^1 \frac{x^k-1}{\log x}dx$$.

The best solution was submitted by Myeongjae Lee (이명재), 2012학번. Congratulations!

Here is his Solution of Problem 2012-22.

Alternative solutions were submitted by 박민재 (2011학번, +3), 서기원 (수리과학과 2009학번, +2), 김태호 (수리과학과 2011학번, +2), 임현진 (물리학과 2010학번, +2), 조위지 (Stanford Univ. 물리학과 박사과정, +3), 박훈민 (대전과학고 2학년, +3).

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# 2012-23 A solution

Prove that for each positive integer $$n$$, there exist $$n$$ real numbers $$x_1,x_2,\ldots,x_n$$ such that $\sum_{j=1}^n \frac{x_j}{1-4(i-j)^2}=1 \text{ for all }i=1,2,\ldots,n$ and $\sum_{j=1}^n x_j=\binom{n+1}{2}.$

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# Solution: 2012-21 Determinant of a random 0-1 matrix

Let $$n$$ be a fixed positive integer and let $$p\in (0,1)$$. Let $$D_n$$ be the determinant of a random $$n\times n$$ 0-1 matrix whose entries are independent identical random variables, each of which is 1 with the probability $$p$$ and 0 with the probability $$1-p$$.  Find the expected value and variance of $$D_n$$.

The best solution was submitted by Myeongjae Lee (이명재), 2012학번. Congratulations!

Here is his Solution of Problem 2012-21.

Alternative solutions were submitted by 박민재 (2011학번, +3), 김태호 (수리과학과 2011학번, +3), 임현진 (물리학과 2010학번, +3), 김지홍 (수리과학과 2007학번, +2), 서기원 (수리과학과 2009학번, +2).

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# Solution: 2012-20 the Inverse of an Upper Triangular Matrix

Let $$A=(a_{ij})$$ be an $$n\times n$$ upper triangular matrix such that $a_{ij}=\binom{n-i+1}{j-i}$ for all $$i\le j$$. Find the inverse matrix of $$A$$.

The best solution was submitted by Minjae Park (박민재), 2011학번. Congratulations!

Here is his Solution of Problem 2012-20.

Alternative solutions were submitted by 서기원 (수리과학과 2009학번, +3), 이명재 (2012학번, +3), 김태호 (수리과학과 2011학번, +3), 임현진 (물리학과 2010학번, +3), 박훈민 (대전과학고 2학년, +3), 윤성철 (홍익대학교 수학교육학과 2009학번, +3), 어수강 (서울대학교 수리과학부 석사과정, +3).

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# 2012-21 Determinant of a random 0-1 matrix

Let $$n$$ be a fixed positive integer and let $$p\in (0,1)$$. Let $$D_n$$ be the determinant of a random $$n\times n$$ 0-1 matrix whose entries are independent identical random variables, each of which is 1 with the probability $$p$$ and 0 with the probability $$1-p$$.  Find the expected value and variance of $$D_n$$.

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Let $$A=(a_{ij})$$ be an $$n\times n$$ upper triangular matrix such that $a_{ij}=\binom{n-i+1}{j-i}$ for all $$i\le j$$. Find the inverse matrix of $$A$$.