Tag Archives: matrix

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 \(x\)?

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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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