IMC 1995 Problem 1

Let $X$ be a invertible matrix with columns $X_{1},X_{2}...,X_{n}$ . Let $Y$ be a matrix with columns $X_{2},X_{3},...,X_{n},0$ . Show that the matrices $A=YX^{-1}$ and $B=X^{-1}Y$ have rank $n-1$ and have only $0$ ´s for eigenvalues.