Simultaneous Linear Equations
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The name of the person in the picture is
20%
A. A$AP Rocky
B. Kid Cudi
C. MC Hammer
D. T.I.
E. Vanilla Ice
A.
20%
B.
20%
20%
C.
D.
20%
E.
4
9
The size of matrix 
5
A.
B.
C.
D.
6
7
2
6
3
7
8
4 is
8
3 4
25%
25%
25%
2.
3.
25%
4 3
3 3
4 4
1.
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4.
10
The c32 entity of the matrix
7
8 
4.1 61
[C ]   9
2
3
4 
 5 6.3 7.2 8.9
A.
B.
C.
D.
25%
25%
25%
2.
3.
25%
2
3
6.3
does not exist
1.
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4.
10
Given
3
[ A]  
5
6
9
2
3
2
[B]  
 8
then if [C]=[A]+[B], c12=
6
9.2
3
6
33%
33%
33%
1. 0
2. 6
3. 12
1.
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2.
3.
10
A square matrix [A] is lower triangular if
A.
B.
C.
D.
aij  0, i  j
25%
25%
25%
2.
3.
25%
aij  0, j  i
aij  0, i  j
aij  0, j  i
1.
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4.
10
An identity matrix [I] needs to satisfy the
following
A.
B.
C.
D.
I ij  0, i  j
25%
25%
25%
2.
3.
25%
I ij  1, i  j
matrix is square
all of the above
1.
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4.
10
4  6 3 
4 3 




Given A  1 2  8, B  9 7 
6  5  9
4  5
then if [C]=[A][B], then c31=
A.
B.
C.
D.
.
-57
-45
57
Does not exist
25%
1.
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25%
2.
25%
3.
25%
4.
10
The following system of equations
x + y=2
6x + 6y=12
has
solution(s).
1.
2.
3.
4.
no
one
more than one but finite number
of
infinite
25%
1.
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25%
25%
2.
3.
25%
4.
10
PHYSICAL PROBLEMS
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Truss Problem
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Pressure vessel problem
c2
u1  c1 r 
r
a
c
b
c4
u 2  c3 r 
r
a
b
4.2857  10 7

7
4
.
2857

10


 6.5

0

 9.2307  10 5
 5.4619  10 5
0
 4.2857  10 7
 0.15384
0
6.5
4.2857  10 7
  c1   7.887  10 3 
0
  

5.4619  10 5  c 2  
0


0.15384  c3   0.007 

 
5 
c
 3.6057  10   4  
0

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Polynomial Regression
We are to fit the data to the polynomial regression model
(T1 ,α1 ) ,(T2 ,α 2 ), ...,(Tn-1 ,α n-1 ) ,(Tn ,α n )
α  a 0  a1T  a 2T 2

 n


 n
  Ti 
 i n1 
 T 2 
i
 
 i 1 
 n

  Ti 
 i 1 
 n 2
  Ti 
 i 1 
 n 3
  Ti 
 i 1 
n
 n 2 

  Ti  
i


 i 1   a   i 1
0
n
n



a    T 
3
T
  i   1 

i
i

i 1
 i 1    
a2   n
n


2
4

T
  Ti  

i i
 i 1
 i 1  
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







END
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Simultaneous Linear Equations
Gaussian Elimination
(Naïve and the Not That So Innocent Also)
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The goal of forward elimination steps in
Naïve Gauss elimination method is to
reduce the coefficient matrix to a (an)
_________
matrix.
1.
2.
3.
4.
diagonal
identity
lower triangular
upper triangular
25%
1.
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25%
25%
2.
3.
25%
4.
One of the pitfalls of Naïve Gauss
Elimination method is
large truncation error
large round-off error
not able to solve
equations with a
noninvertible coefficient
matrix
0%
..
to
bl
e
ta
no
ro
u
rg
e
la
so
.
nd
-o
f..
at
i..
tr
un
c
rg
e
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0%
.
.
0%
la
1.
2.
3.
Increasing the precision of numbers from
single to double in the Naïve Gaussian
elimination method
1.
2.
3.
avoids division by zero
decreases round-off error
allows equations with a
noninvertible coefficient matrix to
be solved
33%
1
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33%
2
33%
3
Division by zero during forward elimination
steps in Naïve Gaussian elimination for
[A][X]=[C] implies the coefficient matrix [A]
1.
2.
3.
is invertible
is not invertible
cannot be determined to be invertible
or not
33%
1.
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33%
2.
33%
3.
Division by zero during forward elimination
steps in Gaussian elimination with partial
pivoting of the set of equations [A][X]=[C]
implies the coefficient matrix [A]
1.
2.
3.
is invertible
is not invertible
cannot be determined to be invertible
or not
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33%
1.
33%
2.
33%
3.
Using 3 significant digit with chopping
at all stages, the result for the following
calculation is
6.095  3.456  1.99
x1 
8
A.
B.
C.
D.
25%
25%
25%
B.
C.
25%
-0.0988
-0.0978
-0.0969
-0.0962
A.
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D.
Using 3 significant digits with rounding-off
at all stages, the result for the following
calculation is
6.095  3.456  1.99
x1 
8
A.
B.
C.
D.
25%
25%
25%
B.
C.
25%
-0.0988
-0.0978
-0.0969
-0.0962
A.
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D.
Determinants
If a multiple of one row of [A]nxn is added or subtracted
to another row of [A]nxn to result in [B]nxn then
det(A)=det(B)
The determinant of an upper triangular matrix
[A]nxn is given by det A  a11  a22  ...  aii  ...  ann  a
n
ii
i 1
Using forward elimination to transform [A]nxn to an
upper triangular matrix, [U]nxn.
Ann  U  nn
det  A  det U 
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Simultaneous Linear Equations
LU Decomposition
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You thought you have parking problems.
Frank Ocean is scared to park when
__________ is around.
25%
25%
25%
B.
C.
25%
A. A$AP Rocky
B. Adele
C. Chris Brown
D. Hillary Clinton
A.
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D.
Truss Problem
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If you have n equations and n unknowns,
the computation time for forward
substitution is approximately proportional
to
33%
A. 4n
B. 4n2
C. 4n3
33%
33%
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A.
B.
C.
If you have a nxn matrix, the computation
time for decomposing the matrix to LU is
approximately proportional to
33%
A. 8n/3
B. 8n2/3
C. 8n3/3
33%
33%
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A.
B.
C.
LU decomposition method is computationally more
efficient than Naïve Gauss elimination for solving
A.
B.
C.
a single set of simultaneous linear
equations
multiple sets of simultaneous linear
equations with different coefficient
matrices and same right hand side
vectors.
multiple sets of simultaneous linear
equations with same coefficient
matrix and different right hand side
vectors
33%
1.
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33%
2.
33%
3.
For a given 1700 x 1700 matrix [A], assume that it takes
about 16 seconds to find the inverse of [A] by the use of
the [L][U] decomposition method. Now you try to use
the Gaussian Elimination method to accomplish the
same task. It will now take approximately ____
seconds.
25%
A.
B.
C.
D.
25%
25%
2
3
25%
4
64
6800
27200
1
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4
For a given 1700 x 1700 matrix [A], assume that it takes
about 16 seconds to find the inverse of [A] by the use of
the [L][U] decomposition method. The approximate
time in seconds that all the forward substitutions take
out of the 16 seconds is
25%
A.
B.
C.
D.
25%
25%
2
3
25%
4
6
8
12
1
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4
THE END
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Consider there are only two computer companies in a country. The companies
are named Dude and Imac. Each year, company Dude keeps 1/5th of its
customers, while the rest switch to Imac. Each year, Imac keeps 1/3rd of its
customers, while the rest switch to Dude. If in 2003, Dude had 1/6th of the
market and Imac had 5/6th of the marker, what will be share of Dude
computers when the market becomes stable?
25%
1.
2.
3.
4.
25%
25%
2.
3.
25%
37/90
5/11
6/11
53/90
1.
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4.
You know Lady Gaga; Who is Shady Gaga
A. Lady Gaga’s sister
B. A person who looks bad with
their sunglasses on
C. A person who looks good with
sunglasses but bad once he/she
takes the sunglasses off
D. That is what Alejandro calls
Lady Gaga
25%
A.
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25%
25%
B.
C.
25%
D.
10
1 0 0 
Given A  0 1 0 


0 0 1.01
then [A] is a
A.
B.
C.
D.
25%
25%
25%
2.
3.
25%
matrix.
diagonal
identity
lower triangular
upper triangular
1.
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4.
A square matrix
[ A]nn
is diagonally dominant if
n
1.
2.
3.
4.
5.
aii   aij , i  1,2,....., n
j 1
i j
aii 
aii 
n
n
a
j 1
i j
ij
n
a
j 1
i j
ij
, i  1,2,....., n and
, for any i  1,2,...., n
aii   aij , i  1,2,....., n and
j 1
20%
20%
20%
20%
20%
n
aii   aij , for any i  1,2,...., n
j 1
n
aii   aij , i  1,2,....., n
j 1
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1.
2.
3.
4.
5.
The following data is given for the velocity of the rocket as a function of time.
To find the velocity at t=21s, you are asked to use a quadratic polynomial
v(t)=at2+bt+c to approximate the velocity profile.
t
(s)
0
14
v
m/s
0
227.04 362.78 517.35
A.
B.
C.
D.
176 14 1 a  227.04
225 15 1 b   362.78

  

400 20 1  c  517.35
225
400

900
15
 0
225


400
0
15
20
 400
 900

1225
20 1 a  517.35
30 1 b   602.97
35 1  c  901.67 
15
20
30
35
602.97 901.67
25%
25%
25%
2.
3.
25%
1 a  362.78
20 1 b   517.35
30 1  c  602.97
1 a   0 
b   362.78
1
  





1  c  517.35

1.
4.
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An example of upper triangular matrix is
A.
B.
C.
D.
 2 3 5
0 0 6


0 0 3
2
0

0
2
6

0
3
0
2
3
2
2
25%
25%
25%
2.
3.
25%
5
6
3
5
3
3
none of the above
1.
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4.
An example of lower triangular matrix is
A.
B.
C.
D.
 2 0 0
 3 0 0


4 5 6
2
3

4
9
2
3

9
5
2
5
6
0
25%
25%
25%
2.
3.
25%
0
0
6
0
0
0
none of the above
1.
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4.
Three kids-Jim, Corey and David receive an inheritance of $2,253,453. The money is
put in three trusts but is not divided equally to begin with. Corey’s trust is three times
that of David’s because Corey made and A in Dr.Kaw’s class. Each trust is put in and
interest generating investment. The total interest of all the three trusts combined at the
end of the first year is $190,740.57 . The equations to find the trust money of Jim (J),
Corey (C) and David (D) in matrix form is
A.
B.
C.
D.
 1
 0

0.06
1
3
0.08
 1
 0

0.06
1
1
0.08
1
0


6
1
1
0


6
1
1
8
3
8
1   J   2,253,453 

 1  C   
0

0.011  D  190,740.57
25%
25%
25%
2.
3.
25%
1   J   2,253,453 

 3  C   
0

0.011  D  190,740.57
1   J   2,253,453 
C   

 3
0
  

11 

 D
 
190,740.57 

1   J   2,253,453 
C   

 1
0
  

11 

 D
 
190,740.57 

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1.
4.
Is how much you are loaded up related to test
score?
Test Score vs Hours of Effort Expected
100
Test Score1
80
60
40
y = -0.0402x + 72.843
R2 = 0.0027
20
0
20
40
60
Hours of Effort Expected
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80
100
Final Grade vs Test#1 Grade
Final Grade vs Test#1 Grade
100
Final Grade
80
60
y = 0.5574x + 36.662
R2 = 0.2783
40
40
50
60
70
80
Test#1 Grade
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90
100
Determinants
If a multiple of one row of [A]nxn is added or subtracted
to another row of [A]nxn to result in [B]nxn then
det(A)=det(B)
The determinant of an upper triangular matrix
[A]nxn is given by det A  a11  a22  ...  aii  ...  ann  a
n
ii
i 1
Using forward elimination to transform [A]nxn to an
upper triangular matrix, [U]nxn.
Ann  U  nn
det  A  det U 
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The name of the person in the picture is
A. Yung Joc
B. Kid Cudi
C. T.I.
D. MC Hammer
25%
A.
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25%
25%
B.
C.
25%
D.
Kanye West is a genius except
A. He grabbed Taylor Swift’s
mike at the VMAs
B. Has diamonds drilled to his
bottom teeth
C. Sings about Mama’s
boyfriend
D. All of the above
25%
A.
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25%
25%
B.
C.
25%
D.
Example of a Poem
Boom Boom Pow,
That is how
I feel when I come to class,
Glad that I have a lot of mass.
I need to integrate my work and life,
Differentiate between love and strife,
Interpolate when my friend whines,
Isn’t that same as reading between the lines?
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This Kiss – Faith Hill
It's a feeling like this
It's centrifugal motion
It's perpetual bliss
It's that pivotal moment
It's, ah unthinkable
This kiss, this kiss
Unsinkable
This kiss, this kiss
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Given
3
[ A]  
5
6
9
2
3
2
[B]  
 8
then if [C]=[A]-[B], c23=
6
9.2
33%
3
6
33%
33%
A. -3
B. 3
C. 9
1.
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2.
3.
A square matrix [A] is upper triangular if
1.
2.
3.
4.
aij  0, i  j
25%
25%
25%
2.
3.
25%
aij  0, j  i
aij  0, i  j
aij  0, j  i
1.
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4.