MATH 307
FALL 2002
EXAM 2
1. (30 points) Let


−3 1 4
A =  −3 1 3 
−1 1 2
The characteristic polynomial of A is fA (λ) = λ3 −3λ+2. Find all eigenvalues and eigenvectors of A, including the algebraic and geometric multiplicity
in each case. Is A diagonalizable?
2. (25 points) Find a basis for the kernel and image of
1 1 2 3
3
A=
2 2 1 0 −3
12
and state the dimension of each. Is ~b =
in the image?
−17
3. (15 points) Let

1

A= k
0

2 0
3 0
0 4
For what values of k are the eigenvalues of A all real?
4. (30 points) True or false? Explain your answer.
a) If A, B are n × n matrices and A is singular then AB is singular.
b) If A is invertible then it is diagonalizable.
c) If ~v is an eigenvector of A then it is also an eigenvector of A 2 + I.
d) If vectors ~v1 , ~v2 , ~v3 span a subspace V of Rn then V has dimension 3.
0 1
1 0
have the same eigenvalues.
and
e) The two matrices
0 1
1 0
f) The following vectors form a basis of R3 :
     
2
5
1
0 7 4
0
0
4