Tag Archives: determinant

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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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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