NOTE: These are only answers to the problems and not full solutions! On the Fnal exam
you will be expected to show all steps used to obtain your answer.
1.
Short Answer Problems
a) det
A
=3

(

2) = 5 so
A

1
=
1
5
±
3

2
11
²
.
b)
L
1
=

100
0
1 0
0
0 1
,L
2
=
1 0
0
0 1
0
00

1
,
so
L
2
◦
L
1
=

1 0
0
010

1
c) Since
C
is 5
×
5 the characteristic polynomial of
C
will be a degree 5 polynomial and
hence must have at least one real root. Thus, it has at least one real eigenvalue which will
give a corresponding real eigenvector.
d) Let
±
b
be in the columnspace of
A
. Then there exist an
±x
∈
R
n
such that
A±x
=
±
b
. But
then
A
±
b
=
A
(
A±x
)=
A
2
=0
=
±
0
.
Hence,
±
b
is in the nullspace of
A
.
2.
a) We rowreduce
A
to get
10
01
b)±rom our rowoperations in a) we have
A
=
1
0 0

510
0
0 1
201
060
001
.
3.
Since
C
is square and the nullspace is a subset of
R
4
,
C
must be 4
×
4. Since the
nullspace of
A
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 Winter '08
 All
 Linear Algebra, Det, cn vn, ∈ Rn

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