MathDB
IMC 2001 Problem 10

Source: IMC 2001 Day 2 Problem 4

October 30, 2020
linear algebramatrix

Problem Statement

Let A=(ak,l)k,l=1,...,nA=(a_{k,l})_{k,l=1,...,n} be a complex n×nn \times n matrix such that for each m∈{1,2,...,n}m \in \{1,2,...,n\} and 1≤j1<...<jm1 \leq j_{1} <...<j_{m} the determinant of the matrix (ajk,jl)k,l=1,...,n(a_{j_{k},j_{l}})_{k,l=1,...,n} is zero. Prove that An=0A^{n}=0 and that there exists a permutation σ∈Sn\sigma \in S_{n} such that the matrix (aσ(k),σ(l))k,l=1,...,n(a_{\sigma(k),\sigma(l)})_{k,l=1,...,n} has all of its nonzero elements above the diagonal.
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