Math 369 HW #8
Due at the beginning of class on Friday, October 31
Reading: Sections DM, SD.
Problems:
1. Let


2 2 0
A = 2 2 0 .
0 0 4
(a) Find numbers a0 , a1 , a2 (not all zero) so that
a0 13 + a1 A + a2 A2 = 0.
(b) Determine the minimal polynomial minpolyA (x) of A (don’t forget to show that it has
minimal degree!).
2. Suppose B is an n × n matrix and let s ∈ R be any number. Prove that
*
*
*
{ x ∈ Rn : B x = s x}
is a subspace of Rn .
3. Let A be the same matrix as in problem #1.
(a) Final all numbers s so that minpolyA (s) = 0.
(b) For each s from part (a), determine
*
*
*
Vs = { v ∈ R3 : A v = s v }.
Find a basis for Vs (by Problem 2, you know it’s a subspace of R3 , so it must have a
basis).
4. Repeat problems 1 and 3 with the matrix


1 −12 e
1
0 .
B = 0
0
0
1
1