LINEAR ALGEBRA 2568 HOMEWORK PROBLEM
YU TSUMURA
Abstract. These are part of the homework assignment due 11/30
Wednesday. Hint: all solutions are somewhere in my website
http://yutsumura.com
Problem 1. Suppose the following information is known about a 3 × 3
matrix A.
 
 
 
 
 
 
1
1
1
1
2
1
A 2 = 6 2 , A −1 = 3 −1 , A −1 = 3 −1 .
1
1
1
1
0
1
(1) Find the eigenvalues of A.
(2) Find the corresponding eigenspaces.
(3) In each of the following questions, you must give a correct reason
(based on the theory of eigenvalues and eigenvectors) to get full
credit. Is A an invertible matrix? Is A an idempotent matrix?
Problem 2. Let A and B be n × n matrices. Suppose that these matrices
have a common eigenvector x. Show that det(AB − BA) = 0.
1
Problem 3. Suppose that
is an eigenvector of a matrix A corresponding
1
2
to the eigenvalue 3 and that
is an eigenvector of A corresponding to the
1
4
eigenvalue −2. Compute A2
.
3
Problem 4. Find a basis for the subspace W of all vectors in R4 which are
perpendicular to the columns of the matrix


11 12 13 14
21 22 23 24

A=
31 32 33 34 .
41 42 43 44
1