Homework 5 Solution Sketches
Math 371, Fall 2011
Assigned:
Sunday, October 9, 201
1
Due:
Thursday, October 20, 2011
(1)
(Pivoting)
(a) Prove that the matrix
±
0 1
1 1
²
does not have an
LU
decomposition.
Use direct decomposition. Suppose that
±
0 1
1 1
²
=
±
1
0
ü
21
1
²±
u
11
u
12
0
u
22
²
=
±
u
11
u
12
ü
21
u
11
ü
21
u
12
+
u
22
²
.
Then
u
11
= 0. But this implies 1 =
ü
21
u
11
= 0.
(b) Does the system
±
0 1
1 1
² ±
x
y
²
=
±
a
b
²
have a unique solution for all
a,b
∈
R
? (Why?)
Yes. The matrix is nonsingular, since
det
±
0 1
1 1
²
=

1
.
(c) How can you modify the system in part (b) so that
LU
decomposition applies?
Switch the two rows.
(2)
(Gaussian Elimination with Partial Pivoting)
Do page 169, # 7(a) by hand.
r
=
1
2
3
2 3 1

4
4 1 4
9
3 4 6
0
r
=
2
1
3
0
5
/
2

1

17
/
2
4
1
4
9
0 13
/
4
3

27
/
4
r
=
2
3
1
0
0

43
/
13

43
/
13
4
1
4
9
0 13
/
4
3

27
/
4
1
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(3)
(Implementing Gaussian Elimination)
Below is a program that computes the
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 Fall '08
 KRASNY
 Math, Gaussian Elimination, Matrix decomposition, mindex

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