MathDB

3

Part of 2003 IMC

Problems(2)

Idempotent

Source: IMC 2003 day 1 problem 3

10/14/2005
Let A∈Rn×nA\in\mathbb{R}^{n\times n} such that 3A3=A2+A+I3A^3=A^2+A+I. Show that the sequence AkA^k converges to an idempotent matrix. (idempotent: B2=BB^2=B)
algebrapolynomiallinear algebramatrixlinear algebra unsolved
IMC 2003 Problem 9

Source: IMC 2003 Day 2 Problem 3

11/2/2020
Let AA be a closed subset of Rn\mathbb{R}^{n} and let BB be the set of all those points b∈Rnb \in \mathbb{R}^{n} for which there exists exactly one point a0∈Aa_{0}\in A such that ∣a0−b∣=inf⁡a∈A∣a−b∣|a_{0}-b|= \inf_{a\in A}|a-b|. Prove that BB is dense in Rn\mathbb{R}^{n}; that is, the closure of BB is Rn\mathbb{R}^{n}
densetopologyreal analysiscollege contestsIMC