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IMC 1995 Problem 1

Source: IMC 1995

February 18, 2021
linear algebramatrix

Problem Statement

Let XX be a invertible matrix with columns X1,X2...,XnX_{1},X_{2}...,X_{n}. Let YY be a matrix with columns X2,X3,...,Xn,0X_{2},X_{3},...,X_{n},0. Show that the matrices A=YX−1A=YX^{-1} and B=X−1YB=X^{-1}Y have rank n−1n-1 and have only 00´s for eigenvalues.
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