Math 225 Assignment #4Due March 26, 2017

A4.1 Let T : M2×2(R)→M2×2(R) be the linear endomorphism

T (A) =[2 33 4

]A.

(a) Show that T is invertible and find T−1.(b) Find the characteristic polynomial, eigenvalues and eigen-

vectors of T .(c) Is T diagonalizable? If it is, find an ordered basis D of

M2×2(R) such that [T ]D←D is diagonal.(d) Compute

T 2017[1 24 3

].

A4.2 Let P3 be the vector space of polynomials in x of degree at most3 and let T : P3 → P3 be the map given by

T (f(x)) = f(x + 1) + f(x− 1).(a) Show that T is a linear endomorphism.(b) Find the characteristic polynomial, eigenvalues and eigen-

vectors of T .(c) Is T diagonalizable? If it is, find an ordered basis D of P3

such that [T ]D←D is diagonal.A4.3 Let {an : n = 0, 1, 2, ...} be a sequence satisfying

an+2 = 2an+1 + 3an + 5n

for all n ≥ 0 and a0 = a1 = 1. Find a formula for an.A4.4 Let A be a square matrix with characteristic polynomial

x4 + x + 1(a) Show that A is invertible.(b) Find the characteristic polynomial of 3I + 2A.

A4.5 Which of the following statements are true and which are false?If it is true, prove it; if it is false, give a counter-example.(a) The sum of two diagonalizable 2× 2 matrices is also diago-

nalizable.(b) A square matrix A is diagonalizable if and only if AT is

diagonalizable.(c) If A is diagonalizable and the characteristic polynomial of

A is (x− 1)n, A must be the n× n identity matrix.1

2

(d) If two matrices A and B are similar, then A2 and B2 arealso similar.