a
) What is the rank of the matrix of coefficients?
Answer:
The rank is 2. This is clear once we rowreduce the augmented matrix:
1
1
1
1
1
1

1
1
1
1
2

3
2
2
2
(2)

(1)
,
(3)

2(1)
→
1
1
1
1
1
0

2
0
0
0
0

5
0
0
0
(3)

5
/
2(2)
→
1
1
1
1
1
0

2
0
0
0
0
0
0
0
0
b
) Does this system have any solutions?
Answer:
Yes.
c
) If the system has one or more solutions, describe them. Is the solution unique?
Answer:
Any vector of the form (1

y

z,
0
, y, z
)
T
will satifisy the system. The solution is not
unique.
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MATHEMATICAL ECONOMICS EXAM #1, SEPTEMBER 26, 2011
Page 2
3. Consider the set
T
=
{
x
∈
R
3
:
k
x
k
= 1
} ⊂
R
3
.
a
) Is the set
T
open, closed, or neither when considered as a subset of
R
3
?
Answer:
The set is closed.
b
) Prove your answer in part (a).
Answer:
One way to prove this is to note that
f
(
x
) =
k
x
k
is continuous, so if
x
n
→
x
, then
k
x
n
k → k
vx
k
. Since
k
x
n
k
= 1,
k
x
k
= 1 and
x
∈
T
.
If you don’t know that the norm is continuous, use the triangle inequality to show
k
x
n
k 
k
x
k
≤ k
x
n

x
k
, which can be used to show that the norm is continuous.
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