Math 2270 Practice Exam 4
-
University of Utah
Fall 2012
Name:
1<
This is a 50 minute exam. Please show all your work, as a worked
problem is required for full points, and partial credit may be rewarded for
some work in the right direction.
1
1. Cofactor Matrices (15 points)
Calculate the cofactor matrix of A:
A=()
/
2
2. Eigenvalues (20 points)
Find the eigenvalues and the corresponding eigenvectors of
LA
I
-I
-S-A
A=(
—12
5)
(-) (-S )
/
I(
I
(cj
-3)
7
Il V
C
j5
/ I
((J/
3
-
3. Diagonalization (20 points)
Diagonalize the matrix
/1 3 0
A=f 31 0
0 0 —2
i-A 3 0
3 I- C
O-Z-A
=
()
[XL1A_
-
33o)
jc
/1
A
(Zr)
3
()
[0
o
0
I
4. Positive Definite Matrices (10 points)
Prove that if A is positive definite, and B is positive definite, then
A + B is positive definite. (Hint A matrix is positive definite if
xTAx> 0fora11xz
0.)
4
-
or
/f
Tor ci/
5
5. Jordan Form
If a matrix has eigenvalues A
Jordan forms of the matrix?
=
2, 2, 1, 0 what are all the possible
/
oo\
/ QL
0
I
OoQ/
6
Lrj
x
H
E
—0
—
—
N
H
—
ci
c)
U
1—
1>c
-
(
—
F’
I’
0
0
It
—v
cs
>
I
—0
7. Linear Transformations (15 points)
2
(a) Is the transformation T 1R
+
T(vi, v
)
2
I1
—
defined by
1 +v
2v
2
a linear transformation? Explain why or provide a counterexample.
(
utv
(vj-
)
r(/)t
(
Z
+
1
T(
2
(b) Is the transformation T 1R
)
2
T(vi, v
—+
—÷
C
)
(
t
1
R defined by
,v
2
, 3v
1
(2v
)
2
1
1 + 2v
a linear transformation? Explain why or provide a counterexample.
v
T
()
/(2
(i
D
T() t()
+v
1
(L
5 i)
(o/l/