# www.rc.usf.edu Simultaneous Linear Equations http://nm.mathforcollege.com 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. 34 B. 43 C. 33 D. 4 4 6 2 7 3 6 7 8 4 is 8 25%

1. http://nm.mathforcollege.com 25% 25% 2. 3. 25% 4. 10 The c32 entity of the matrix 4.1 61 [C ] 9 2 5 6.3 A. B. C. D. 7 3 7.2

8 4 8.9 25% 25% 25% 2. 3. 25% 2 3 6.3 does not exist 1. http://nm.mathforcollege.com 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.

http://nm.mathforcollege.com 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.

http://nm.mathforcollege.com 4. 10 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 25% 1. http://nm.mathforcollege.com

25% 25% 2. 3. 25% 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. -57 B. -45 C. 57 D. Does not exist 25%

1. http://nm.mathforcollege.com 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% http://nm.mathforcollege.com 1. 25% 25% 2. 3. 25% 4. 10 PHYSICAL PROBLEMS http://nm.mathforcollege.com Truss Problem http://nm.mathforcollege.com Pressure vessel problem c2 u1 c1 r r

a c c4 u 2 c3 r r b 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 3.6057 10 5 c 4 0 http://nm.mathforcollege.com Polynomial Regression We are to fit the data to the polynomial regression model (T1 ,1 ) ,(T2 , 2 ), ...,(Tn-1 , n-1 ) ,(Tn , n ) 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 Ti 3 a1 Ti i i 1 i 1 a2 n n 2 4 T Ti i i i 1 i 1 http://nm.mathforcollege.com END

http://nm.mathforcollege.com Simultaneous Linear Equations Gaussian Elimination (Nave and the Not That So Innocent Also) http://nm.mathforcollege.com The goal of forward elimination steps in Nave Gauss elimination method is to reduce the coefficient matrix to a (an) _________ matrix. 1. 2. 3. 4. diagonal identity lower triangular upper triangular 25% http://nm.mathforcollege.com 1. 25% 25%

2. 3. 25% 4. One of the pitfalls of Nave Gauss Elimination method is 1. 2. 3. large truncation error large round-off error not able to solve equations with a noninvertible coefficient matrix 0% http://nm.mathforcollege.com 0% 0% Increasing the precision of numbers from single to double in the Nave 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 http://nm.mathforcollege.com 33% 2 33% 3 Division by zero during forward elimination steps in Nave 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. http://nm.mathforcollege.com 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 http://nm.mathforcollege.com 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. http://nm.mathforcollege.com 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. http://nm.mathforcollege.com 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 det A a11 a22 ... aii ... ann a matrix [A]nxn is given by n i 1 Using forward elimination to transform [A]nxn to an upper triangular matrix, [U]nxn. A nn U nn det A det U http://nm.mathforcollege.com ii Simultaneous Linear Equations

LU Decomposition http://nm.mathforcollege.com 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. http://nm.mathforcollege.com D. Truss Problem http://nm.mathforcollege.com

If you have n equations and n unknowns, the computation time for forward substitution is approximately proportional to A. 4n B. 4n2 C. 4n3 33% http://nm.mathforcollege.com A. 33% B. 33% 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 http://nm.mathforcollege.com

A. 33% B. 33% C. LU decomposition method is computationally more efficient than Nave Gauss elimination for solving A. B. C. a single set of simultaneous linear equations 33% 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 1. http://nm.mathforcollege.com

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 http://nm.mathforcollege.com 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 A. B. C. D. 4 6 8 12 25% 1 http://nm.mathforcollege.com 25% 25%

2 3 25% 4 THE END http://nm.mathforcollege.com 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. http://nm.mathforcollege.com 4. You know Lady Gaga; Who is Shady Gaga A. Lady Gagas 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% http://nm.mathforcollege.com A. 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. http://nm.mathforcollege.com 4. A square matrix 1. 2. 3. 4. 5. [ A]nn n is diagonally dominant if aii aij , i 1,2,....., n j 1 i j n

aii aij , i 1,2,....., n and j 1 i j aii n n aij , for any i 1,2,...., n j 1 i j 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 http://nm.mathforcollege.com 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 15

20 30 35 v m/s 0 227.0 4 362.7 8 517.3 5 602.9 7 901.6 7 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 20 0 225 400 0 15 400 900 1225 20 1 a 517.35 30 1 b 602.97 35 1 c 901.67 30 20

25% 25% 25% 2. 3. 25% 1 a 362.78 1 b 517.35 1 c 602.97 1 a 0 1 b 362.78 1 c 517.35 1. 4. http://nm.mathforcollege.com An example of upper triangular matrix is A. B. C. D. 2

0 0 3 5 0 6 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. http://nm.mathforcollege.com 4. An example of lower triangular matrix is A. B. C. D. 2 3 4 0 0 5

0 0 6 2 3 4 9 2 5 0 0 6 2 3 9 5 0 0 0 6 0 25%

25% 25% 2. 3. 25% none of the above 1. http://nm.mathforcollege.com 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. Coreys trust is three times that of Davids because Corey made and A in Dr.Kaws 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. 1 0 0.06

1 3 0.08 1 J 2,253,453 1 C 0 0.011 D 190,740.57 B. 1 0 0.06 1 1 1 J 2,253,453 3 C 0 0.011 D 190,740.57 C. 1 0

6 1 1 8 1 J 2,253,453 3 C 0 190,740.57 11 D D. 1 0 6 1 3 8 1 J 2,253,453 1 C 0 190,740.57

11 D 0.08 http://nm.mathforcollege.com 25% 1. 25% 25% 2. 3. 25% 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 http://nm.mathforcollege.com 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 http://nm.mathforcollege.com 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 det A by a11 a22 ... aii ... ann a matrix [A]nxn is given n i 1 Using forward elimination to transform [A]nxn to an upper triangular matrix, [U]nxn.

A nn U nn det A det U http://nm.mathforcollege.com ii The name of the person in the picture is A. Yung Joc B. Kid Cudi C. T.I. D. MC Hammer 25% A. http://nm.mathforcollege.com 25% 25% B. C. 25% D. Kanye West is a genius except A. He grabbed Taylor Swifts

mike at the VMAs B. Has diamonds drilled to his bottom teeth C. Sings about Mamas boyfriend D. All of the above 25% A. http://nm.mathforcollege.com 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,

Isnt that same as reading between the lines? http://nm.mathforcollege.com 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 http://nm.mathforcollege.com 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. http://nm.mathforcollege.com 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. http://nm.mathforcollege.com 4.