#### Transcript www.eng.usf.edu

**Simultaneous Linear Equations**

**http://nm.mathforcollege.com**

The size of matrix 4 9 5 A.

B.

3 4 4 3 C.

D.

3 3

### 4

### 4

6 2 6 7 3 7 8 4 8 is

**25% 25% 25% 25% 1.**

**2.**

**3.**

**4.**

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**10**

The c

*32*

entity of the matrix [

*C*

] 4 .

1 9 5 A.

B.

C.

D.

61 2 6 .

3 2 3 6.3

does not exist 7 3 7 .

2 8 4 8 .

9

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**2.**

**3.**

**4.**

**10**

Given [

*A*

] 3 5 6 9 2 3 [

*B*

then if [C]=[A]+[B], c 12 = ] 2 8 6 9 .

2

**33%**

3 6

**33%**

A. 0 B. 6 C. 12

**33% 1.**

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**2.**

**3.**

**10**

Given [

*A*

] 3 5 6 9 2 3 [ then if [C]=[A]-[B], c 23 =

*B*

] 2 8 9 6 .

2

**33%**

3 6

**33%**

A.

B.

C.

-3 9 3

**33% 2.**

**3.**

Given 4 1 6 6 2 5 3 8 9 , 4 9 4 3 7 5 then if [*C*]*=*[*A*][*B*], then *c*

*31*

= .

**25% 25% 25% 25%**

A.

B.

C.

D.

-57 -45 57 does not exist

**1.**

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**2.**

**3.**

**4.**

**10**

A square matrix [A] is lower triangular if

**25% 25% 25% 25%**

A.

B.

C.

D.

*a a ij*

0 ,

*i*

*a ij a ij ij j*

0 ,

*j*

0 ,

*i*

0 ,

*j*

*j i i*

**1.**

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**2.**

**3.**

**4.**

**10**

A square matrix [A] is upper triangular if

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1.

2.

3.

4.

*a a ij*

0 ,

*i*

*a ij a ij ij j*

0 ,

*j*

0 ,

*i*

0 ,

*j*

*j i i*

**1.**

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**2.**

**3.**

**4.**

An example of upper triangular matrix is A.

B.

C.

D.

2 0 0 2 0 0 2 6 0 5 6 3 5 6 3 5 3 3 3 0 0 3 2 2 3 0 2 none of the above

**25% 25% 25% 25% 2.**

**3.**

**4.**

An example of lower triangular matrix is A.

B.

C.

D.

2 3 4 2 3 4 0 0 5 9 2 5 0 0 6 0 0 6 2 3 9 5 6 0 0 0 0 none of the above

**25% 25% 25% 25% 2.**

**3.**

**4.**

An identity matrix [

*I*

] needs to satisfy the following A.

*I ij*

0 ,

*i*

*j*

B.

*I ij*

1 ,

*i*

*j*

C.

matrix is square D.

all of the above

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**2.**

**3.**

**4.**

**10**

Given 1 0 0 0 1 0 1 .

0 0 01 then *[A] *is a matrix.

A.

B.

C.

D.

diagonal identity lower triangular upper triangular

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**2.**

**3.**

**4.**

A square matrix [

*A*

]

*n*

*n*

is diagonally dominant if 1.

2.

3.

4.

*a ii a ii*

*n*

*i j*

1

*j n*

*i j*

1

*j a ij*

,

*i a ij*

1 , 2 ,.....,

*n*

,

*i*

1 , 2 ,.....,

*n and a ii a ii*

*j n*

1

*n*

*i j*

1

*j a ij*

,

*i a ij*

,

*for any i*

1 , 2 ,.....,

*n and*

1 , 2 ,....,

*n a ii a ii*

*j n*

1

*j n*

1

*a ij*

,

*a ij for any i*

,

*i*

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

*n n*

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**25% 25% 25% 25% 1.**

**2.**

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**4.**

The following system of equations

*x + y=2*

6

*x + *

6

*y=12*

has solution(s).

1.

2.

3.

4.

no one more than one but a finite number of infinite

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**1.**

**2.**

**3.**

**4.**

**10**

**PHYSICAL PROBLEMS http://nm.mathforcollege.com**

Truss Problem

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**Pressure vessel problem**

a b a c

*u*

1

*c*

1

*r*

*c*

2

*r u*

2

*c*

3

*r*

*c*

4

*r*

b 4 .

2857 4 .

2857 6 0 .

5 10 10 7 7 9 .

2307 10 5 5 .

4619 10 5 0 .

15384 0 0 4 .

2857 10 7 6 .

5 4 .

2857 10 7 5 .

4619 0 .

0 10 15384 5 3 .

6057 10 5

*c c*

*c c*

1 2 3 4 7 .

887 0 .

0 10 3 007 0

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### Polynomial Regression

We are to fit the data to the polynomial regression model (

*T*

1

*,α*

1 )

*,*

(

*T*

2

*,α*

2 ),

*...,*

(

*T n-*

1

*,α n-*

1 )

*,*

(

*T n ,α n*

)

*α*

*a*

0

*a*

1

*T*

*a*

2

*T*

2

*i n*

1

*i n*

1

*n T i T i*

2

*i n*

1

*T i*

*i n*

1

*T i*

2

*i n*

1

*T i*

3

*i n*

1

*T i*

2

*i n*

1

*T i*

3

*i n*

1

*T i*

4

*a*

0

*a*

1

*a*

2

*i i n*

1

*n*

1

*i n*

1

*T i T i*

2

*i i i*

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**END http://nm.mathforcollege.com**

**Simultaneous Linear Equations Gaussian Elimination**

**(Naïve and the Not That So Innocent Also) http://nm.mathforcollege.com**

The goal of forward elimination steps in Naïve Gauss elimination method is to reduce the coefficient matrix to a (an) _________ matrix.

A.

B.

C.

D.

diagonal identity lower triangular upper triangular

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**1.**

**2.**

**3.**

**4.**

One of the pitfalls of Naïve Gauss Elimination method is A.

B.

C.

large truncation error large round-off error not able to solve equations with a noninvertible coefficient matrix

**33% 33% 33% http://nm.mathforcollege.com**

**1.**

**2.**

**3.**

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

B.

C.

avoids division by zero decreases round-off error allows equations with a noninvertible coefficient matrix to be solved

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**1 2 3**

Division by zero during forward elimination steps in

*Naïve Gaussian elimination*

for [A][X]=[C] implies the coefficient matrix [A]

**33% 33% 33%**

1.

2.

3.

is invertible is not invertible cannot be determined to be invertible or not

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**1.**

**2.**

**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

**33% 33% 33% http://nm.mathforcollege.com**

**1.**

**2.**

**3.**

Using 3 significant digit with

*chopping *

at all stages, the result for the following calculation is

*x*

1 6 .

095 3 .

456 1 .

99 8

**25% 25% 25% 25%**

A. -0.0988

B. -0.0978

C. -0.0969

D. -0.0962

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**A.**

**B.**

**C.**

**D.**

Using 3 significant digits with

*rounding-off *

at all stages, the result for the following calculation is

*x*

1 6 .

095 3 .

456 1 .

99 8

**25% 25% 25% 25%**

A. -0.0988

B. -0.0978

C. -0.0969

D. -0.0962

**A.**

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**B.**

**C.**

**D.**

**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% 25%**

A. A$AP Rocky B. Adele C. Chris Brown D. Hillary Clinton

**A.**

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**B.**

**C.**

**D.**

Truss Problem

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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*

11

*a*

22 ...

*a ii*

...

*a*

*nn i n*

1

*a ii*

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

*n*

*n*

*n*

*n*

det det to an

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If you have

*n *

equations and

*n *

unknowns, the computation time for forward substitution is approximately proportional to

**33% 33% 33%**

*A. 4n B. 4n*

2

*C. 4n*

3

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**A.**

**B.**

**C.**

If you have a

*n*

x

*n *

matrix, the computation time for decomposing the matrix to LU is approximately proportional to

**33% 33% 33%**

*A. 8n/3 B. 8n*

2 /3

*C. 8n*

3 /3

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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% http://nm.mathforcollege.com**

**1.**

**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% 25% 25%**

A.

4

**25%**

B.

64 C.

D.

6800 27200

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**1 2 3 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% 25% 25% 25%**

A.

4 B.

6 C.

D.

8 12

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**2 3 4**

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)=at 2 +bt+c*

to approximate the velocity profile.

*t v*

(s) m/s 0 0 14 227.04

15 362.78

20 517.35

30 602.97

35 901.67

**25% 25% 25% 25%**

A.

B.

C.

D.

176 225 400 14 15 20 1 1 1

*a b c*

227 .

04 362 .

78 517 .

35 225 400 900 0 225 400 400 900 1225 15 20 30 0 15 20 20 30 35 1 1 1

*a*

*b c*

362 517 602 .

.

78 .

35 97 1 1 1

*a b c*

0 362 517 .

.

78 35 1 1 1

*b a*

*c*

517 602 901 .

.

.

35 97 67

**1.**

**2.**

**3.**

**4.**

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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

**25% 25% 25% 25%**

A.

1 0 0 .

06 1 3 0 .

08 0 .

1 1 011

*J C D*

2 , 253 190 , 0 , 453 740 .

57 B.

C.

0 1 0 .

06 1 1 0 .

08 0 .

1 3 011

*J C D*

2 , 253 190 0 , 453 , 740 .

57 D.

1 0 6 1 1 8 1 3 11

*J C D*

2 , 253 190 0 , 453 , 740 .

57 1 0 6 1 3 8 1 1 11

*J C D*

2 , 253 190 0 , 453 , 740 .

57

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**1.**

**2.**

**3.**

**4.**

**THE END**

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**4.09**

**Adequacy of Solutions**

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The well or ill conditioning of a system of equations [A][X]=[C] depends on the A. coefficient matrix only B. right hand side vector only C. number of unknowns D. coefficient matrix and the right hand side vector

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**1.**

**2.**

**3.**

**4.**

The condition number of a

*n×n*

diagonal matrix [A] is A.

B.

max

*a ii*

,

*i*

1 ,...,

*n*

min max

*a ii a ii*

,

*i*

,

*i*

1 ,...,

*n*

1 ,...,

*n*

2 C.

D.

2 min 1

*a ii*

,

*i*

1 ,...,

*n*

max

*a ii*

,

*i*

1 ,...,

*n*

min

*a ii*

,

*i*

1 ,...,

*n*

**25% 25% 25% 25% http://nm.mathforcollege.com**

**1.**

**2.**

**3.**

**4.**

The adequacy of a simultaneous linear system of equations [A][X]=[C] depends on (choose the most appropriate answer) A. condition number of the coefficient matrix B. machine epsilon C. product of the condition number of coefficient matrix and machine epsilon D. norm of the coefficient matrix

**25% 25% 25% 25% http://nm.mathforcollege.com**

**1.**

**2.**

**3.**

**4.**

If

*cond*

and

*mach*

0 .

119 10 6 , then in [A][X]=[C], at least these many significant digits are correct in your solution,

**25% 25% 25% 25%**

A. 0 B. 1 C. 2 D. 3

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**1.**

**2.**

**3.**

**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 1/5 th of its customers, while the rest switch to

*Imac*

. Each year,

*Imac*

customers, while the rest switch to keeps 1/3

*Dude. *

If in 2003,

*Dude *

had 1/6 th rd of its of the market and

*Imac *

had 5/6 th of the marker, what will be share of

*Dude *

computers when the market becomes stable?

**25% 25% 25% 25%**

1.

2.

3.

4.

37/90 5/11 6/11 53/90

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**1.**

**2.**

**3.**

**4.**