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SAMPLE FINAL 1 ANSWERS
NOTE: These are only answers to the problems and not full solutions! On the ﬁnal 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
1
1
²
.
b)
L
1
=

1 0 0
0
1 0
0
0 1
,
L
2
=
1 0
0
0 1
0
0 0

1
,
so
L
2
◦
L
1
=

1 0
0
0
1
0
0
0

1
c)
λ
is an eigenvalue and
~v
6
=
~
0 is an eigenvector of a matrix
A
if
A~v
=
λ~v
.
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
~x
= 0
~x
=
~
0
.
Hence,
~
b
is in the nullspace of
A
.
2.
a) We rowreduce
A
to get
1 0
0 1
0 0
b)From our rowoperations in a) we have
A
=
1
0 0

5 1 0
0
0 1
1 0 0
0 1 0
2 0 1
1 0 0
0 6 0
0 0 1
1 0
0 1
0 0
.
3.
Since
C
is square and the nullspace is a subset of
R
4
,
C
must be 4
×
4. Since the
nullspace of
A
has one parameter, we must have the rank of
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 Fall '08
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