# 2021-07 Odd determinant

Let $$A_N$$ be an $$N \times N$$ matrix whose entries are i.i.d. Bernoulli random variables with probability $$1/2$$, i.e.,

$\mathbb{P}( (A_N)_{ij} =0) = \mathbb{P}( (A_N)_{ij} =1) = \frac{1}{2}.$

Let $$p_N$$ be the probability that $$\det A_N$$ is odd. Find $$\lim_{N \to \infty} p_N$$.

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# 2017-10 An inequality for determinant

Let $$A$$, $$B$$ be matrices over the reals with $$n$$ rows. Let $$M=\begin{pmatrix}A &B\end{pmatrix}$$. Prove that $\det(M^TM)\le \det(A^TA)\det(B^TB).$

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# 2016-9 Determinant of a matrix

Let $$A=(a_{ij})_{ij}$$ be an $$n\times n$$ matrix, where $a_{ij}=\begin{cases} 3 &\text{if }i=j,\\ (-1)^{\lvert i-j\rvert}&\text{otherwise.}\end{cases}$ Compute the determinant of $$A$$.

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# 2016-4 Distances in a tree

Let $$T$$ be a tree on $$n$$ vertices $$V=\{1,2,\ldots,n\}$$. For two vertices $$i$$ and $$j$$, let $$d_{ij}$$ be the distance between $$i$$ and $$j$$, that is the number of edges in the unique path from $$i$$ to $$j$$. Let $$D_T(x)=(x^{d_{ij}})_{i,j\in V}$$ be the $$n\times n$$ matrix. Prove that $\det (D_T(x))=(1-x^2)^{n-1}.$

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# 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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# 2014-05 Nonnegative determinant

Let $$n$$, $$k$$ be positive integers and let $$A_1,A_2,\ldots,A_n$$ be $$k\times k$$ real matrices. Prove or disprove that $\det\left(\sum_{i=1}^n A_i^t A_i\right)\ge 0.$  (Here, $$A^t$$ denotes the transpose of the matrix $$A$$.)

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# 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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# 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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