B U Department of Mathematics
Math 201 Matrix Theory
Spring 2006 Final Exam
This archive is a property of Bo˘gazi¸ci University Mathematics Department. The purpose of this archive is to organise and centralise the distribution of the exam questions and their solutions.
This archive is a nonprofit service and it must remain so. Do not let anyone sell and do not buy this archive, or any portion of it. Reproduction or distribution of this archive, or any portion of
it, without nonprofit purpose may result in severe civil and criminal penalties.
1.
Prove: If the entries in each row of an
nxn
matrix A add up to zero, then
det
(
A
) = 0. (Hint:
Consider the product AX where X is an nx1 matrix, each of whose entries is one) (20 points)
Solution:
For
A
=
⎡
⎢
⎣
a
11
a
12
· · ·
a
1
n
.
.
.
a
n
1
a
n
2
· · ·
a
nn
⎤
⎥
⎦
and
X
=
⎡
⎢
⎢
⎢
⎣
1
1
.
.
.
1
⎤
⎥
⎥
⎥
⎦
AX
=
⎡
⎢
⎢
⎢
⎣
a
11
+
a
12
+
· · ·
+
a
1
n
a
21
+
a
22
+
· · ·
+
a
2
n
.
.
.
a
n
1
+
a
n
2
+
· · ·
+
a
nn
⎤
⎥
⎥
⎥
⎦
=
⎡
⎢
⎢
⎢
⎣
0
0
.
.
.
0
⎤
⎥
⎥
⎥
⎦
(Since the entries in each row of A add up to zero)
So
X
=
⎡
⎢
⎢
⎢
⎣
1
1
.
.
.
1
⎤
⎥
⎥
⎥
⎦
= 0 is a solution of
AX
= 0
⇒
det
(
A
) = 0.
(Recall: If
det
(
A
) = 0, then
AX
= 0 has only the trivial solution,
X
= 0).
2.
Find the equation of the best line through the points (1,2), (0,0), (1,1) and (2,3). (25 points)
Solution:
Let
y
=
C
+
Dt
be the best line fitting the given data.
Then for
A
=
⎡
⎢
⎢
⎣
1
−
1
1
0
1
1
1
2
⎤
⎥
⎥
⎦
,
b
=
⎡
⎢
⎢
⎣
−
2
0
1
3
⎤
⎥
⎥
⎦
and
X
=
C
D
(
A
T
A
)
x
=
A
T
b
must hold.
A
T
A
=
1
1
1
1
−
1
0
1
2
⎡
⎢
⎢
⎣
1
−
1
1
0
1
1
1
2
⎤
⎥
⎥
⎦
=
4
2
2
6
A
T
b
=
1
1
1
1
−
1
0
1
2
⎡
⎢
⎢
⎣
−
2
0
1
3
⎤
⎥
⎥
⎦
=
2
9
⇒
4
2
2
6
C
D
=
2
9
i
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
⇒
4
C
+ 2
D
= 2
2
C
+ 6
D
= 9
⇒
D
= 1
.
6
, C
=
−
0
.
3
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 SOYSAL
 Math, Linear Algebra, Matrices, Bo˘azi¸i University Mathematics Department

Click to edit the document details