MA 353, Practice Midterm 2
March 29, 2010
Name:
This exam consists of 6 pages including this front page.
Ground Rules
(1) No calculator is allowed.
(2) Show your work for every problem unless stated otherwise. A correct an
swer without any justification may not receive the full credit.
(3) You may use one 4by6 index card, both sides.
(4) Show your ID on your desk.
Score
1
10
2
10
3
10
4
10
5
10
Total
50
1
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2
1.
Let
A
=
1
2
0
0
1

1
0
0
2
.
(a) Find all the eigenvalues of
A
and their geometric and algebraic multi
plicities
Answer.
Consider
det(
A

λI
3
) =
1

λ
2
0
0
1

λ

1
0
0
2

λ
= (1

λ
)
2
(2

λ
) = 0
which gives
λ
= 1
,
2 as the eigenvalues with algebraic multiplicities
2 and 1, respectively. Clearly for
λ
= 2 the geometric multiplicity is
1 since the algebraic multiplicity is 1. So it remains to compute the
geometric multiplicity for
λ
= 2. Recall that the geometric multiplicity
of
λ
= 1 is dim
N
(
A

I
3
). The matrix
A

I
3
reduces to
0
1
0
0
0
1
0
0
0
,
whose rank is 2, and so the nullity (
i.e.
the dimension of the null
space) is 1. Hence the geometric multiplicity of
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 Spring '08
 Staff
 Linear Algebra, Algebra, geometric multiplicity, algebraic multiplicities

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