5. Recall that for an n x n matrix A, its characteristic polynomial PA(x) is defined by PA(x) := det(xIn – A), where In denotes the n x n identity matrix. Consider the sequence of simple graphs Tn =...

Extracted text: 5. Recall that for an n x n matrix A, its characteristic polynomial PA(x) is defined by PA(x) := det(xIn – A), where In denotes the n x n identity matrix. Consider the sequence of simple graphs Tn = (Vn, En) defined as follows for n 2 0: Vn := {vo, . .., Vn} and En := {{vi-1, vi} | i = 1, ...,n}, with the convention that E, = Ø. Page 2 of 3 (a) Find the adjacency matrix An of the graph Tn for each n > 0. (b) Set Po(x) := 1, and for n > 1 let Pn(x) := det(xIm – An-1) be the characteristic polynomial of A,–1. Find P(x) and P2(x). (c) Show that for n > 2 we have that Pn(x) = xPn-1(x) – Pn-2(x). (Hint for part (c): consider the Laplace expansion of the determinant with respect to the first column of xI, – An-1-)