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MAE 107 – Computational Methods
Summer Session II 2009
Midterm – Solutions
Wednesday, August 19
4 problems + 5 short questions – 40 points
Problem 1 – 10 points
Consider the matrix
A
=
3
2
0

3

3 2
6
4
4
.
1. The ﬁrst step of the Forward Elimination procedure aims at canceling the coeﬃcients below the pivot
coeﬃcient
a
11
= 3 in the ﬁrst column of
A
. To do so, we deﬁne
λ
21
=
a
21
a
11
=

1 and
λ
31
=
a
31
a
11
= 2.
The following row operations are then performed:

(2)
←
(2)

λ
21
(1)
⇔
(2) + (1)

(3)
←
(3)

λ
31
(1)
⇔
(3)

2
×
(1)
The result of this operation can be written as a decomposition for
A
:
A
=
1
0 0

1 1 0
2
0 1
3
2
0
0

1 2
0
0
4
We observe that the ﬁrst step cancelled all the subdiagonal coeﬃcients and therefore no second step
is necessary to remove the subdiagonal coeﬃcient of the second column. The
λ
ij
are the subdiagonal
coeﬃcients of
L
. The LU decomposition of
A
is therefore simply:
A
=
L
·
U
,
with
L
=
1
0 0

1 1 0
2
0 1
and
U
=
3
2
0
0

1 2
0
0
4
.
2. Using the fundamental property of the determinant

A

=

LU

=

L
·
U

.
L
and
U
are triangular
matrices: their determinant is simply the product of their diagonal coeﬃcients.

L

= 1 and

U

=

12, therefore

A

=

12
.
Using the deﬁnition of the determinant and expanding over the ﬁrst row of
A
(since it has a zero
coeﬃcient):

A

=3
×
±
±
±
±

3 2
4
4
±
±
±
±

2
×
±
±
±
±

3 2
6
4
±
±
±
±
+ 0
×
±
±
±
±

3

3
6
4
±
±
±
±
=3(

12

8)

2(

12

12)
=

60 + 48
=

12

A
 6
= 0 therefore the square system
Ax
=
b
has a unique solution.
1
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View Full Document3. To compute
Y
=
A

1
we need to solve 3 linear systems for the column vectors
y
(1)
,
y
(2)
and
y
(3)
,
which are the columns of
Y
:
Ay
(1)
=
1
0
0
,
Ay
(2)
=
0
1
0
,
Ay
(3)
=
0
0
1
.
Using the LU decomposition of
A
we can replace these three systems by three pairs of triangular
systems:
±
Lz
(
k
)
=
b
Uy
(
k
)
=
z
(
k
)
for 1
≤
k
≤
3
.
For each
k
= 1, 2 and 3, we can solve successively
Lz
(
k
)
=
b
for
z
(
k
)
using Forward Substitution
and
Uy
(
k
)
=
z
(
k
)
using Backward Substitution.

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 Summer '08
 Rottman

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