King Fahd University of Petroleum and Minerals
Department of Mathematics and Statistics
(Math 280)
Code: A
Final Exam
Term 141
Tuesday, Jan 06, 2015
Building No. 04 (Room No. 101)
Net Time Allowed: 180 minutes
07:00PM to 10:00PM
Name:
ID:
(SHOW ALL YOUR STEPS AND WORK)
PART-I
Time Allowed:
hours
True2-1/2
/ FALSE
PART-II
WRITTEN - PART
/20
Q
1
Points
/16
2
/16
3
/16
4
/16
5
/16
Total
PART-III
/80
MCQ - PART
Q
Answers
Points
1
a
b
c
d
e
2
a
b
c
d
e
/4
/4
3
a
b
c
d
e
/4
4
a
b
c
d
e
/4
5
a
b
c
d
e
/4
6
a
b
c
d
e
/4
7
a
b
c
d
e
/4
8
a
b
c
d
e
/4
9
a
b
c
d
e
/4
10
a
b
c
d
e
/4
Total
/40
Grand Total
/140
PART-I
Please mark the following statements with True or False.
1)

Let A be an n  n matrix. If det A  0 , then A1  
1 
adj A .
T 
 det A 
T
2)
F
An n  n matrix is orthogonal if AT A  I
T
5)
F
An n  n matrix A can be orthogonally diagonalized if and only if A is
symmetric.
/02
T
4)
F
An n  n matrix A is diagonalizable if and only if A has n linearly
independent eigenvectors .
/02
T
3)
/02
F
The set cos5x, sin 5x is a linearly dependent set.
T
/02
F
/02
6)
Let S1 and S2 be finite subsets of a vector space and let S1 be a subset of
/02
S 2 . Then, if S1 is linearly dependent, then S 2 is linearly independent.
T
7)
A line through the origin of R3 is a subspace of R3 .
T
8)
F
F
Elementary row operations do not change the row space of a matrix.
T
10)
/02
A finite set of vectors that contains the zero vectors is linearly independent.
/02
T
9)
F
F
For a square 2  2 matrix A , if A 2  A, then    1 are eigenvalues of A .
T
/02
F
/02
PART-II
1) Given the vectors u  4, 1
I)
a)
and
v  2, 3
/16
Find a unit vector w in the direction of v .
b) Find the angle between u and
v.
c) Find the projection of u onto v.
II)
Prove that if a, b and c are vectors such that c is orthogonal to both
a and b , then c is orthogonal to every vector in span a, b
2) Consider the basis S  u1 , u2 , u3  for the Euclidean space R3 with the
standard inner product where
/16
 1
 1 
 1 
 
 
 
u1  1 , u2   0  , and u3   2 
 1
 1 
 3
 
 
 
By using Gram-Schmidt process, Transform S to an orthonormal basis
T  v1 , v2 , v3  .
 9 1 1
3) Given the Symmetric matrix A  1 9 1 .
1 1 9 


i) Compute the eigenvalues of A .
ii)
Compute the corresponding eigenvectors.
iii)
Find the Matrix X that, ORTHOGONALLY, diagonalize A.
Also, find the diagonal Matrix D.
/16
4) Given the linear Transformation L : R 3  R 3
 x1   x1  x3
  
L   x2    x1  x2
 x   2x  x
 3  1
2


 2 x3 

 3x3 
a) Find the representation matrix A .
b) Find the ker of L.
c) Find the Rank of A .
L  x   A x defined by
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5) Use Cramer’s rule to solve the system
3 x1  2 x2  x3  7
x1  x2  3x3  3
5 x1  4 x2  2 x3  1
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PART-III
1 3 1 
3 2 1 


1) If A   2 0 1  , B  1  4 1  and 2 X  3A  X  B , then the sum of the
1 0 0
0 2 4
elements in the main diagonal of the matrix X is equal to:
A)
8
B)
-8
C)
-4
D)
10
E)
0
2) If the matrix
1
0
A
0

0
0
0
1
1 1
0
0
2
1
0
0 2
1

1
1

2 
is the augmented matrix of a
linear system whose solution is ( a, b, c, d), then a+b+c+d =
A)
0
B)
4
C)
-4
D)
3
E)
-3
/04
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3) The value of k for which the system with augmented matrix
1
0

0
2
3
1
2
0
k 1


1
 has infinitely many solutions is:
2
k  3k  4 
0
A)
is equal to
4
B)
is equal to
1
C)
is equal to
1
D)
does not exist
E)
is equal to
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0
4) If A is a 3  3 square matrix with A  4 , and the matrix B is obtained
from the matrix A by interchanging the first and the last rows, then the
1
value of 2 A  8 B
is equal to
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A) 34
B)
30
C)
10
D)
6
E)
68
5  2

5) For the matrix  2 4
 4  5
 3
1  the sum of the minor of the element  5 and
6
the cofactor of the element 2 is
A)
65
B)
- 38
C)
- 16
D)
16
E)
38
 2
 1
6) Let A  
AX B
3
, and
2
2 
x 
B    , and X    . If the solution of the system
 y
3 
is X  CB , then C 
 2
7
A) 
 1
7

 3 
7
 2 
7
 2
7
B) 
 1
7

 3 
7
 2 
7
 2
7
C) 
 1
7

3 
7
2 
7
 2
7
D) 
 1
7

 3 
7
2 
7
 1
E)  2
 1

1 
3
1 
2
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1 0 1 


7) The Sum of all elements of the inverse matrix of 0 1 1  is
0 0 3
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4
3
A)
B)

2
3
C)

1
3
D)
2
3
E)
5
3
8) If the system of equations
2x  3y  1
3x  4 y  3
x  ky  2
is consistent, then the value of the constant k is
A)
2
B)
1
C)
1
3
D)
1
E)
2
3
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9) Given the vectors u  4, 4
and v  15, 18 . If the vector w  m, n
is a unit vector in the opposite direction of
1
1
u
v , then m  n 
2
3
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1
5
A)
B)

1
5
C)

7
5
D)
7
5
E)
2
5
a
10) If
A)
48
B)
–6
C)
– 12
D)
12
E)
24
b
c
d e
f  3,
g h
k
then
2d
2e
2f
2a
 2b
 2c
2 g  2a
2h  2b
2k  2c

/04