Math 205B Quiz 07 page 1 03/12/2010 Name
1. Suppose that M=0k
−1 6 .
1A. Find all values of kfor which Mhas λ= 3 as an eigenvalue with multiplicity 2. Show all your work.
1B. Find all values of kfor which Mhas eigenvalues 5 and 1. Show all your work.
1C. Find all values of kfor which Mhas no real eigenvalues. Show all your work.
2. Let A=
8−3 2
2 1 4
−5 15 −3
. Here are three facts about A: Fact 1) One eigenvalue of Ais α= 7.
Fact 2) The vector v=
1
2
−5
is an eigenvector of Afor some other eigenvalue β6= 7.
Fact 3) In no part of this question do you need to use determinants.
2A. Use fact 2 to find that other eigenvalue β. Show all your work.
2B. Find a basis for the eigenspace for the eigenvalue 7. Show all your work.
2C. Find the characteristic polynomial of A.