Tag Archives: matrix

2015-18 Determinant

What is the determinant of the n×n matrix An=(aij) where aij={1,if i=j,x,if |i−j|=1,0,otherwise, for a real number x?

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2013-01 Inequality involving eigenvalues and traces

Let A,B be N×N symmetric matrices with eigenvalues λA1≤λA2≤⋯≤λAN and λB1≤λB2≤⋯≤λBN. Prove that

\sum_{i=1}^N |\lambda_i^A – \lambda_i^B|^2 \leq Tr (A-B)^2

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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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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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2012-9 Rank of a matrix

Let M be an n⨉n matrix over the reals. Prove that \operatorname{rank} M=\operatorname{rank} M^2 if and only if \lim_{\lambda\to 0}  (M+\lambda I)^{-1}M exists.

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2012-6 Matrix modulo p

Let p be a prime number and let n be a positive integer. Let A=\left( \binom{i+j-2}{i-1}\right)_{1\le i\le p^n, 1\le j\le p^n} be a p^n \times p^n matrix. Prove that A^3 \equiv I \pmod p, where I is the p^n \times p^n identity matrix.

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2011-14 Invertible matrices

For a positive integer n>1, let f(n) be the largest real number such that for every n×n diagonal matrix M with positive diagonal entries, if tr(M)<f(n), then M-J is invertible. Determine f(n). (The matrix J is the square matrix with all entries 1.)

(Due to a mistake, the problem is fixed at 3:30PM Friday.)

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