Math 2270 Final
-
University of Utah
Fall 2012
Name:
1<
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This is a 2 hour exam. P1e 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. Elimination and LUFactorization
(a) Use elimination to calculate the row echelon form of the matrix
/ 1
A=( —1
2
7
—2 3\
3 0)
—5 Sj
3 \
-
-L3
(13
‘
7
(1
L row(
I
1
o
5
) 7i
( o
ij\
J
\\O O2//
2
(b) Calculate the LU factorization of the matrix A from part (a) of
this problem, given below
1
—1
2
4-II
—2 3
3 0
—5 5
uI
j
55
(c) Is the matrix A invertible? How do you know?
1+ h
,
oi1-e,a,
3
i/
C’-.
0
C)
II
I
—
cI:
o
U
C)
—
—
Ct
‘-
I
0
—.
r.J
Th
0
C)
ct
II
0
U
C)
Ct
0
C)
C)
0
L
J
Q}
-J
-
H
0
0
11
3. Elminination, Rank, and Complete Solutions
(a) Use elimination to calculate the row echelon form of the matrix
‘2 4 6 4\
2 5 7 6
B=
(
9
5b47c
Iromrow
L 5
$
rcc
YOcU I rciv
(
5L)
1
0
1 1
0 -1 H
)
2 3 5 2j
4
cj
I
L
)
L/
z
I)
-L
0000/
5
(b) What is the rank of the matrix B from part (a) if this problem?
q re
s
L
(c) What are the dimensions of the following subspaces for the ma
trix B:
• Row space C(BT)
• Nullspace N(B)
LZl
-
Z
-
• Column space C(B)
LZ1
-
• Left nullspace N(BT)
-
6
3
Z
e
(d) What is a basis for the nulispace, N(B), of the matrix B from part
(a) of this problem?
6L)
(o/cui111J
00
cA
IlL
000
XO
1//
0
XI
yl
3
(
-i
\o
/
L
Lj
6
I
L)
0’
o
/
0
C)
(0
/
5
7-I
I
(
C)
7
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(e) Find the complete solution to the system of equations
4r
1 + 4x
2x
2 + 6x
3 + 4
4
1 + 5x
2x
2 + 7x
3 + 6x
1
2x
+
2
3x
+
3
5x
+
4
2x
4
5
3
=
=
=
£°
/1
J
L/ L/
I
4co
1
row
L
row3
+c
)
o
11
cj
?_
,
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A.
,
(c
EQ
*0
xL
14
co/
j•?
1/o
0
8
Nl(pce
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4. Projections
Calculate the projection of the vector a onto the vector b:
a= —31
br=
2i)
p roJt
-±
(0)
++J7
3
3)
rIdj
()
9
2
21
5. Eigenvalues and Diagonalization
(a) Calculate the eigenvalues of the matrix
/1 0
0 —l
G=
—2
(
2 \
—2
0
J
i-
1
C
-‘
O
,
3
qr
10
i
1
a
S
S
C
—
ii
9>s
>
—
I
I
K
11
lx
11
>c
FHH —H
0
-
0
(c) Diagoflalize, C
(
=
1
S
,
AS the matrix C from part (a).
to
a -H
2,
C) Li
(3
-L
C
‘2,
3
0•
0
i
3 0
0 0 3
3
3
0 0
I
3
z
-3
j
2,
3
3
3
3
3
3
3
T
L-Z
(d) Is the matrix C positivedeftmt Why or why not?
4
12
/i
0
ç
jive
•
-
L
—
x
a)
C
a)
a)
—
b
a)
a)
U
ou
-s
4-
cJThLr
4
J
_
H
c
—
K
c)
Um
IN
.—.- - -
U\
—\
fj
(b) Calculate the pseudoinverse of the matrix D from part (a).
(
z
j2 V
14
7. Graphs and Incidence Matrices
(a) What is the incidence matrix for the graph:
I
3
5
IOo\
I -1
I-I
0
(0
(0
(0-1
\o-i a)
\Oo 1 —(
15
(b) Perform elimination on the indicence matrix from part (a).
—1
I00
(Ho 0
(01
0.-I I 0
3 -(0/
C) 0/-
6Ydw)jQ
a
5±ictc*row/
/
f-)
/
/)
/
00
0 00
OHIO
—I (0
0
\o c
0-1(o)
C)ec)
0c-l
(0
0
113
C)
0
1o
0OC)C)
1
I
0006)
(c) What is the spanning tre for the graph from part
by the reduction of the indicence matrix calculated in part (b)?
16
0
8. Markov Matrices
(a) What are the eigenvalues of the Markov matrix
M=(
:)
Hint It’s not necessary to find the roots of the characteristic
equation! Just use what you know about eigenvalues of positive
Markov matrices, and how the sum of the eigenvalues relates to
the trace of the matrix.
-
_3
H
15
2
(Y
-
17
1<0 V
a+r1
(b) What is the steady-state solution for the Markov matrix ii from
part (a)?
efl
coY
5
-5 -7
0
/?
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18
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