Linear Algebra S2010D Sec.1, Summer 2006
Final Exam
Name:
June 27, 2006
Do all problems, in any order.
Show your work. An answer alone may not receive full credit.
No notes, texts, or calculators may be used on this exam.
Problem
Possible
Points
Points
Earned
1
15
2
10
3
15
4
15
5
10
6
10
7
15
8
10
TOTAL
100
1
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1.
(15 points) Given
A
=
3
2
2
3
(a) Explain what it means for a symmetric matrix
B
to be
positive definite
.
(b) Is
A
positive definite? Explain.
(c) Find a matrix
B
such that
B
2
=
A
or explain why no such matrix exists (you need not
simplify your answer).
(d) Find a matrix
C
such that
e
C
=
A
or explain why no such matrix exists (you need not
simplify your answer).
2.
(10 points) Recall that a square matrix
A
is called
nilpotent
if there exists a positive
integer
n
such that
A
n
= 0.
(a) Show that the eigenvalues of a nilpotent matrix are zero.
(c) Give an example of a nonzero nilpotent matrix.
3.
(15 points) Given the singular value decomposition
A
=
U
Σ
V
T
with
U
=
1
3

1
2
2
2

1
2
2
2

1
Σ =
4
0
0
0
0
1
0
0
0
0
0
0
V
=
1
2
1
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 Spring '10
 Peters
 Linear Algebra, Singular value decomposition, Orthogonal matrix, γ

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