Math 2270 Exam 1
-
University of Utah
Fall 2012
Name:
(ey
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. (15 points) Vector Basics
For the vectors
arrr( 1)
b=f i)
4J
\1)
/1
c=f 2
answer the following, or explain why the question does not make
sense:
(a) (3 points) 2a + 3c
i(
=
(D
(‘)f()
1
i)
4)
arrf
b=(
ii
/1
c=( 2
1J
(b) (3 points) aH
l+I6
(c) (2 points) What are the components of a unit vector in the same
direction as a?
/
/
2
c)
N
cD.
N
0
Q
II
+
H
r
T
0
I.
II
I’
I
2. (10 points) Matrix Basics
For the matrices
(3 4 2
A
2
1 1
B=
/2 1 5
4 4 2
1 02
(
1 1
0 0
answer the following, or explain why the question does not make
sense:
(a) (3points)A+Crr
(3
-
ttj
16oO
4
cDCC
CDCN
II
(z
C
r\rJ
—
C
Cj
c)
0
0
‘-Th
1
c’-
Lfl
3. (15 points) Elimination Issues
(a) (5 points) For what value of a in the system of equations below
does elimination fail to produce a unique solution?
3x + 2y
6x + ay
=
=
ir
11
/c{
Ji1 (
a e
C
10
b
rc ie4
j
1
ru
Z
1/
/ce;
q
h-
/e
p
O
Oy
(b) (5 points) Given the determined value of a, for what value of b
are there an infinite number of solutions?
T
b?o
7(j
o Oy
a(ty1
s
(c) (5 points) For the determined values of a and b what are two dis
tinct solutions?
/
0
y
Thte
/
qfN/
o4t
6
(QL/J
4. (20 points) Systems of Equations
Use elementary row operations to convert the system of equations
2r
6r
12x
+
+
+
3y +
6y +
9y
—
3z
=
12z
z
=
3
13
2
into upper-triangular form, and then use back-substitution to solve
for the variables x, y, z. Be sure to show all your work.
5
(Rc
1Q-/
5
L +
t1
Z
Y
3
fc(J3
Z\4Jy 3y
LI
y
3
-
rc, /J
(f or/
7
x
a)
—c
—QQ
I
oç
—
—
—
—
—
-
c_J
_
,
H
C
a)
—
0
——H
_Lf
—
a)
-d
co
I
----
—
11
JZ
cx-)
c)
><
ct
0)
C
0
U
J —r
0
UN
Ui
fl
7. (10 points) Symmetric Products
For the matrix
2 3
4 0
(a) (4 points) What is the transpose RT?
(b) (4 points) What is the symmetric product RTR
73
1
LC)
30/
(c) (2 points) Does RTR
=
RRT?
/
he
10