# 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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# 2015-4 An inequality on positive semidefinite matrices

Let $$M=\begin{pmatrix} A & B \\ B^*& C \end{pmatrix}$$ be a positive semidefinite Hermian matrix. Prove that $\operatorname{rank} M \le \operatorname{rank} A +\operatorname{rank} C.$ (Here, $$A$$, $$B$$, $$C$$ are matrices.)

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# 2013-01 Inequality involving eigenvalues and traces

Let $$A, B$$ be $$N \times N$$ symmetric matrices with eigenvalues $$\lambda_1^A \leq \lambda_2^A \leq \cdots \leq \lambda_N^A$$ and $$\lambda_1^B \leq \lambda_2^B \leq \cdots \leq \lambda_N^B$$. 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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# 2012-20 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 $$A$$.

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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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# 2011-15 Two matrices

Let n be a positive integer. Let ω=cos(2π/n)+i sin(2π/n). Suppose that A, B are two complex square matrices such that AB=ω BA. Prove that (A+B)n=An+Bn.

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