simultaneous linear equations

Simultaneous Linear Equations

http://nm.mathforcollege.com

The name of the person in the picture is

A. B. C. D. E.

20% 20% 20%20%20%

A. A$AP Rocky

B. Kid Cudi

C. MC Hammer

D. T.I.

E. Vanilla Ice

The size of matrix

43

1. 2. 3. 4.

25% 25%25%25%

34

33

44

A.

B.

C.

D.

8765

4329

8764

is

10http://nm.mathforcollege.com

The c32 entity of the matrix

1. 2. 3. 4.

25% 25%25%25%

9.82.73.65

4329

87611.4

][C

A. 2B. 3

C. 6.3

D. does not exist


Given

1. 2. 3.

33% 33%33%

395

263][A

62.98

362][B

then if [C]=[A]+[B], c12=

1. 0

2. 6

3. 12


A square matrix [A] is lower triangular if

jiaij ,0

1. 2. 3. 4.

25% 25%25%25%

ijaij ,0

jiaij ,0ijaij ,0

A.

B.

C.

D.


An identity matrix [I] needs to satisfy the following

jiI ij ,0

1. 2. 3. 4.

25% 25%25%25%

jiI ij ,1

A.

B.

C.

D. all of the above

matrix is square


Given

1. 2. 3. 4.

25% 25%25%25%

then if [C]=[A][B], then c31= .A. -57

B. -45

C. 57

D. Does not exist


54

79

34

,

956

821

364

BA

The following system of equations x + y=26x + 6y=12has solution(s).

1. 2. 3. 4.

25% 25%25%25%1. no2. one3. more than one but finite number

of4. infinite


PHYSICAL PROBLEMS


Truss Problem


Pressure vessel problem

a

b

a

b

c

0

007.0

0

10887.7

106057.3102857.400

15384.05.615384.05.6

104619.5102857.4104619.5102857.4

00102307.9102857.4 3

4

3

2

1

57

5757

57

c

c

c

c

r

crcu 2

11

r

crcu 4

32


Polynomial Regression

2210 TaTaaα

We are to fit the data to the polynomial regression model

n

iii

n

iii

n

ii

n

ii

n

ii

n

ii

n

ii

n

ii

n

ii

n

ii

n

ii

T

T

a

a

a

TTT

TTT

TTn

1

2

1

1

2

1

0

1

4

1

3

1

2

1

3

1

2

1

1

2

1

)()(),()( 112211 nnn-n- ,αT,,αT...,,αT,,αT


END



Gaussian Elimination(Naïve and the Not That So Innocent Also)


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.

25% 25%25%25%

1. diagonal

2. identity

3. lower triangular

4. upper triangular


One of the pitfalls of Naïve Gauss Elimination method is

0% 0%0%

1. large truncation error

2. large round-off error

3. not able to solve equations with a noninvertible coefficient matrix


Increasing the precision of numbers from single to double in the Naïve Gaussian elimination method

1 2 3

33% 33%33%

1. avoids division by zero

2. decreases round-off error

3. allows equations with a noninvertible coefficient matrix to be solved


Division by zero during forward elimination steps in Naïve Gaussian elimination for [A][X]=[C] implies the coefficient matrix [A]

1. 2. 3.

33% 33%33%1. is invertible

2. is not invertible

3. cannot be determined to be invertible or not


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.

33% 33%33%1. is invertible

2. is not invertible

3. cannot be determined to be invertible or not


Using 3 significant digit with chopping at all stages, the result for the following calculation is

A. B. C. D.

25% 25%25%25%8

99.1456.3095.61

x

A. -0.0988

B. -0.0978

C. -0.0969

D. -0.0962


Using 3 significant digits with rounding-off at all stages, the result for the following calculation is

A. B. C. D.

25% 25%25%25%

8

99.1456.3095.61

x

A. -0.0988

B. -0.0978

C. -0.0969

D. -0.0962


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 nnii aaaa ......Adet 2211

n

iiia

1

Using forward elimination to transform [A]nxn to an upper triangular matrix, [U]nxn.

nnnn UA

UA detdet http://nm.mathforcollege.com


LU Decomposition


You thought you have parking problems. Frank Ocean is scared to park when __________ is around.

A. B. C. D.

25% 25%25%25%

A. A$AP Rocky

B. Adele

C. Chris Brown

D. Hillary Clinton


Truss Problem


If you have n equations and n unknowns, the computation time for forward substitution is approximately proportional to

A. B. C.

33% 33%33%A. 4n

B. 4n2

C. 4n3


If you have a nxn matrix, the computation time for decomposing the matrix to LU is approximately proportional to

A. B. C.

33% 33%33%A. 8n/3

B. 8n2/3

C. 8n3/3


LU decomposition method is computationally more efficient than Naïve Gauss elimination for solving

1. 2. 3.

33% 33%33%

A. a single set of simultaneous linear equations

B. multiple sets of simultaneous linear equations with different coefficient matrices and same right hand side vectors.

C. multiple sets of simultaneous linear equations with same coefficient matrix and different right hand side vectors


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.

1 2 3 4

25% 25%25%25%

A. 4

B. 64

C. 6800

D. 27200


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

1 2 3 4

25% 25%25%25%A. 4

B. 6

C. 8

D. 12


THE END


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?

1. 2. 3. 4.

25% 25%25%25%

1. 37/902. 5/113. 6/114. 53/90


You know Lady Gaga; Who is Shady Gaga

A. B. C. D.

25% 25%25%25%

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


Given

1. 2. 3. 4.

25% 25%25%25%

01.100

010

001

A

then [A] is a matrix.

A. diagonal

B. identity

C. lower triangular

D. upper triangular


A square matrix nnA ][

1. 2. 3. 4. 5.

20% 20% 20%20%20%

is diagonally dominant if

niaan

jij

ijii ,.....,2,1,1

n

jij

ijii

n

jij

ijii

nianyforaa

andniaa

1

1

,....,2,1 ,

,.....,2,1,

n

jijii

n

jijii

nianyforaa

andniaa

1

1

,....,2,1 ,

,.....,2,1,

niaan

jijii ,.....,2,1,

1

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.

35.517

78.362

04.227

120400

115225

114176

c

b

a

1. 2. 3. 4.

25% 25%25%25%

97.602

35.517

78.362

130900

120400

115225

c

b

a

35.517

78.362

0

120400

115225

100

c

b

a

67.901

97.602

35.517

1351225

130900

120400

c

b

a

t (s) 0 14 15 20 30 35

v m/s 0 227.04

362.78

517.35

602.97

901.67

A.

B.

C.

D.


An example of upper triangular matrix is

300

600

532

1. 2. 3. 4.

25% 25%25%25%

320

600

532

320

326

532

A.

B.

C.

D. none of the above


An example of lower triangular matrix is

654

003

002

1. 2. 3. 4.

25% 25%25%25%

654

023

092

009

063

052

A.

B.

C.

D. none of the above


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

57.740,190

0

453,253,2

011.008.006.0

310

111

D

C

J

1. 2. 3. 4.

25% 25%25%25%

57.740,190

0

453,253,2

011.008.006.0

130

111

D

C

J

57.740,190

0

453,253,2

1186

310

111

D

C

J

57.740,190

0

453,253,2

1186

130

111

D

C

J

A.

B.

C.

D.


Is how much you are loaded up related to test score?

Test Score vs Hours of Effort Expected

y = -0.0402x + 72.843

R2 = 0.002720

40

60

80

100

0 20 40 60 80 100

Hours of Effort Expected

Tes

t S

core

1


Final Grade vs Test#1 GradeFinal Grade vs Test#1 Grade

y = 0.5574x + 36.662

R2 = 0.2783

40

60

80

100

40 50 60 70 80 90 100

Test#1 Grade

Fin

al G

rade


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 nnii aaaa ......Adet 2211

n

iiia

1

Using forward elimination to transform [A]nxn to an upper triangular matrix, [U]nxn.

nnnn UA

UA detdet http://nm.mathforcollege.com

The name of the person in the picture is

A. B. C. D.

25% 25%25%25%

A. Yung Joc

B. Kid Cudi

C. T.I.

D. MC Hammer


Kanye West is a genius except

A. B. C. D.

25% 25%25%25%

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


Example of a PoemBoom 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?


This Kiss – Faith Hill

It's a feeling like thisIt's centrifugal motionIt's perpetual blissIt's that pivotal momentIt's, ah unthinkableThis kiss, this kissUnsinkableThis kiss, this kiss


Given

1. 2. 3.

33% 33%33%

395

263][A

62.98

362][B

then if [C]=[A]-[B], c23=

A. -3

B. 3

C. 9


A square matrix [A] is upper triangular if

jiaij ,0

1. 2. 3. 4.

25% 25%25%25%

ijaij ,0

jiaij ,0ijaij ,0

1.

2.

3.

4.


simultaneous linear equations

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