Math x + 3y 5z = 14 3x 2y + 3z = 17 4x + 3y 2z = 1
|
|
|
- Juniper West
- 5 years ago
- Views:
Transcription
1 Math Solve the system: x + y + z = 1 2x + 3y + 4z = 5 (a z = 2, y = 1 and x = 0 (b z =any value, y = 3 2z and x = z 2 (c z =any value, y = 3 2z and x = z + 2 (d z =any value, y = 3 + 2z and x = z 2 (e None of the above 2. Solve the system: In the solution x =: (a 4 (b 3 (c 1 (d 2 (e The system is inconsistent. 2x + 3y 5z = 14 3x 2y + 3z = 17 4x + 3y 2z = 1 3. Solve the system: 5x 2y 2z = 5 2x y z = 1 3x + y + 2z = 2 (a In the solution x = 4 (b In the solution x = 2 (c In the solution x = 1 (d In the solution x = 3 (e None of the above 1
2 4. Find the bottom left entry of: ( ( (a 22 (b 48 (c 43 (d 64 (e Solve the system: x + y 2z = 3 2x 3y + 3z = 2 5x 5y + 4z = 6 (a In the solution x = 4. (b In the solution x = 2. (c In the solution x = 3. (d In the solution x = 1. (e The system is inconsistent. 6. Find the inverse of A = The entry in the first row and first column of A 1 is: (a The matrix A is not invertible. (b 1/2. (c -5/4. (d 0. (e 3/4. 2
3 7. Which of the following systems have infinitely many solutions? { x y + 8z + 3w = 10 x y + 8z = 7 I. II. y 8z = 3 y 8z = 3 w = 1 III. (a I, II, and IV only. (b IV only. (c I, II, and III only. (d I and II only. x y + 8z = 7 y 8z = 3 y 8z = 3 IV. x y + 5z = 7 y 8z = 3 y 8z = 4 8. Invert the following matrix In the inverted matrix, the top right entry is: (a 65 (b 80 (c 5 (d -30 (e Find the bottom left entry of: ( ( (a 23 (b 19 (c 34 (d 64 (e 48 3
4 10. Which of the following matrices are invertible? ( ( A =, B = (a A and D only. (b A only. (c A and B only. (d All except C. (e Some other selection., C = ( ( 0 1, D = A = Solve the system: In the solution: (a w=0. (b w=1. (c w=3. (d w= A 1 = x + 2y + 3z + 4w = 0 2x + 5y + 10z + 20w = 1 3x + 7y + 14z + 21w = 2 4x + 9y + 17z + 26w = Assume that we have a simple economy with only two industries Gas and Electricity. To produce 1 unit of Gas requires.1 units Gas and.1 units of Electricity. To produce 1 unit of Electricity requires.4 units Gas and.4 units of Electricity. There is a consumer demand for 25 units of Gas and 15 units of Electricity. How many units of gas should be produced? (a 57 (b 42 (c 37 (d 25 (e 30 4
5 13. A simple economy has 2 industries: oil and gas. The production of $1 of oil requires $0.10 in oil and $0.30 in gas, while the production of $1 of gas requires $0.20 of oil and $0.40 of gas. Assume there is an external demand for $240 of oil and $480 of gas. How much oil should be produced to satisfy the demand? (a $1050. (b $500. (c $750. (d $ If A = ( then the top right entry of A + B is: (a 1. (b 3. (c 0. (d 4. (e 2. and B = ( In solving a system of linear equations using Gauss-Jordan elimination, the following augmented matrix is obtained: Which is true? x y z (a There are exactly two solutions for this problem. (b There are infinitely many solutions, one of which is x = 0, y = 11, z = 2. (c There is exactly one solution and x = 6, y = 7, and z = 0. (d There are infinitely many solutions, one of which is x = 9, y = 5, z = 1. (e There are no solutions for this problem. 5
6 16. Solve the system : x y + 2z = 14 y 2z = 7 y + 2z = 33 (a In the solution: x = 1. (b There are infinitely many solutions. (c No solution; the equations are inconsistent. (d In the solution: x = 6. (e In the solution: x = Suppose that the matrix product BAB is defined, and is Then: (a The size of B is 7 7. (b The size of B is impossible to determine. (c BAB can never have size (d The size of B is (e The size of B is Solve the system: x + 2y + z = 8 3x + 7y + 4z = 29 3x 3y 5z = 18 (a In the solution x = 3. (b In the solution x = 2. (c In the solution x = 4. (d In the solution x = 1. (e The system is inconsistent. 6
7 19. Given that the solution of (a 6 (b 4 (c 14 (d 10 (e No solution exists = x + 4z = 8 5x +y +8z 2w = 7 6x 2y +z = 4 x 4z +w = 2, what is the z value in 20. Assume that we have a simple economy with only two industries Gas and Electricity. To produce 1 unit of Gas requires.52 units Gas and.44 units of Electricity. To produce 1 unit of Electricity requires.25 units Gas and.25 units of Electricity. There is a consumer demand for 32 units of Gas and 54 units of Electricity. How many units of gas should be produced? (a 160 (b 250 (c 128 (d 32 (e Which of the following matrices are invertible? ( ( A =, B = (a A and D only. (b A only (c A and B only. (d All except C. (e None., C = ( ( 0 1, D = 0 1 7
8 22. Solve the system : x + 3y z = 1 x + 4y z = 4 2x + 7y 2z = 5 (a There are infinitely many solutions. (b In the solution: y = 2. (c No solution; the equations are inconsistent. (d In the solution: y = 0. (e In the solution: y = Find the inverse of The top left entry of B 1 is: (a 2 (b 3 (c 1 (d 4 B = A simple economy has 2 industries: oil and gas. The production of $1 of oil requires $0.10 in oil and $0.30 in gas, while the production of $1 of gas requires $0.20 of oil and $0.40 of gas. Assume there is an external demand for $240 of oil and $480 of gas. How much gas should be produced to satisfy the demand? (a $1050. (b $1300. (c $500. (d $750. 8
9 25. Which of the following matrices are invertible? ( ( ( A =, B =, C = (a A and B only. (b A only. (c B only. (d All except C. (e All of them. ( 1 0, D = If [ x (a 6 (b 15 (c 12 (d 18 (e 20 ] is not invertible, what is x? 27. Solve the system : In the solution: (a z = 2. (b z = 1. (c z = 0. (d There are infinitely many solutions. x 2y + 5z = 1 2x + 3y 5z = 5 3x + 2y + 5z = 5 (e No solution; the equations are inconsistent. 9
10 28. Assume that we have a simple economy with only two industries Gas and Electricity. To produce 1 unit of Gas requires.1 units Gas and.1 units of Electricity. To produce 1 unit of Electricity requires.4 units Gas and.4 units of Electricity. There is a consumer demand for 25 units of Gas and 15 units of Electricity. How many units of electricity should be produced? (a 15 (b 27 (c 32 (d 23 (e Let A be a 4 3 matrix, B a 3 5 matrix, and C a 4 5 matrix. Which of the following is defined? (a CB T + A T (b A T C + B (c All of the above. (d AB + C T 30. If A = then the top right entry of AB is: (a 0 (b 1 (c 3 (d 2 (e -1 ( and B =
11 31. Find the top left entry of: ( ( (a 7 (b 5 (c 1 (d Find the inverse of Which of the following is true? (a The top right entry of the inverse is 0. (b The top left entry of the inverse is 1. (c The matrix has more than one inverse. (d The matrix is not invertible (e The bottom right entry of the inverse is 1/ A simple economy has 2 industries: oil and gas. The production of $1 of oil requires $0.10 in oil and $0.20 cents in gas, while the production of $1 of gas requires $0.30 cents of oil and $0.40 cents of gas. Assume there is an external demand for $960 of oil and $1920 of gas. How much oil should be produced to satisfy the demand? (a $2000. (b $4000 (c $1200. (d $
12 34. Invert the following matrix and then answer the question In the inverted matrix, the entry in the middle row and middle column is: (a 3 (b 0 (c 2 (d 1 (e None of the above 35. Invert the following matrix In the inverted matrix, the top right entry is: (a 4 (b -4 (c 0 (d -1 (e Let A = column of BA is (a 0 (b 20 (c -24 (d 4 (e 3 and B = [ ]. The entry in the first row, second 12
13 37. Which of the following systems have infinitely many solutions? { x y + 5z + 3w = 13 x y + 5z = 10 I. II. x y = 3 x y = 3 w = 1 III. (a I, II, III, and IV. (b I, II, and III only. (c I and II only. (d III and IV only. (e IV only. x y + 5z = 10 x y = 3 x y = 3 IV. x y + 5z = 10 x y = 3 x + y = Assume that we have a simple economy with only two industries Gas and Electricity. To produce 1 unit of Gas requires.10 units Gas and.14 units of Electricity. To produce 1 unit of Electricity requires.50 units Gas and.70 units of Electricity. There is a consumer demand for 100 units of Gas and 200 units of Electricity. How many units of gas should be produced? (a 600. (b 650. (c 500. (d (e Solve the system : x y + 5z = 13 y 2z = 7 y + 8z = 33 (a In the solution: x = 3. (b No solution; the equations are inconsistent. (c In the solution: x = 1. (d In the solution: x = 6. (e There are infinitely many solutions. 13
14 40. Invert the following matrix and then answer the question In the inverted matrix the entry in the top row and middle column is: (a 2 (b 1 (c 3 (d 4 (e None of the above 41. Find the inverse of A = (a 1. (b The entry in the second row and second column of A 1 is: 0. (c The matrix A is not invertible. (d 1/2. (e -1/2. 14
15 42. Consider the system: Given: A = In the solution: (a w=2. (b w=0. (c w=1. (d w= x + 2y + 3z + 5w = 0 x + 3y + 8z + 4w = 1 2x + 5y + 12z + 25w = 2 4x + 10y + 23z + 35w = 3 A 1 = Find the inverse of The entry in the third row and third column is: (a 1/7. (b 0. (c -1. (d 1. 15
16 44. Solve the system : In the solution: (a There are infinitely many solutions. (b z = 0. (c z = 1. x + y + 10z = 12 2x + 3y 5z = 5 3x + 2y + 5z = 5 (d No solution; the equations are inconsistent. (e z = Invert the following matrix In the inverted matrix the top right entry is? (a 4 (b 0 (c -4 (d -1 (e Solve the system: In the solution x =: (a 4 (b 2 (c 3 (d 1 (e The system is inconsistent. x + y 2z = 3 2x 3y + 3z = 2 5x 10y + 11z = 7 16
17 47. Suppose that the matrix product ABC is defined where A is 2 5 and C is 7 9. Then B is : (a 5 7. (b 5 9. (c 7 5. (d 2 9. (e A simple economy has 2 industries: oil and gas. The production of $1 of oil requires $0.10 in oil and $0.20 cents in gas, while the production of $1 of gas requires $0.30 cents of oil and $0.40 cents of gas. Assume there is an external demand for $960 of oil and $1920 of gas. How much gas should be produced to satisfy the demand? (a $1200. (b $4000 (c $2000. (d $1000. bddceccdcb - 10 bbbddeedde - 20 bacacdccba - 30 bdeaeebbdd - 40 bedceeab 17
Chapter 6 Page 1 of 10. Lecture Guide. Math College Algebra Chapter 6. to accompany. College Algebra by Julie Miller
Chapter 6 Page 1 of 10 Lecture Guide Math 105 - College Algebra Chapter 6 to accompany College Algebra by Julie Miller Corresponding Lecture Videos can be found at Prepared by Stephen Toner & Nichole DuBal
MATH 1003 Review: Part 2. Matrices. MATH 1003 Review: Part 2. Matrices
Matrices (Ch.4) (i) System of linear equations in 2 variables (L.5, Ch4.1) Find solutions by graphing Supply and demand curve (ii) Basic ideas about Matrices (L.6, Ch4.2) To know a matrix Row operation
Section 1.1 System of Linear Equations. Dr. Abdulla Eid. College of Science. MATHS 211: Linear Algebra
Section 1.1 System of Linear Equations College of Science MATHS 211: Linear Algebra (University of Bahrain) Linear System 1 / 33 Goals:. 1 Define system of linear equations and their solutions. 2 To represent
If A is a 4 6 matrix and B is a 6 3 matrix then the dimension of AB is A. 4 6 B. 6 6 C. 4 3 D. 3 4 E. Undefined
Question 1 If A is a 4 6 matrix and B is a 6 3 matrix then the dimension of AB is A. 4 6 B. 6 6 C. 4 3 D. 3 4 E. Undefined Quang T. Bach Math 18 October 18, 2017 1 / 17 Question 2 1 2 Let A = 3 4 1 2 3
Matrices (Ch.4) MATH 1003 Review: Part 2. Matrices. Matrices (Ch.4) Matrices (Ch.4)
Matrices (Ch.4) Maosheng Xiong Department of Mathematics, HKUST (i) System of linear equations in 2 variables (L.5, Ch4.) Find solutions by graphing Supply and demand curve (ii) Basic ideas about Matrices
Linear Algebra Practice Problems
Math 7, Professor Ramras Linear Algebra Practice Problems () Consider the following system of linear equations in the variables x, y, and z, in which the constants a and b are real numbers. x y + z = a
Methods for Solving Linear Systems Part 2
Methods for Solving Linear Systems Part 2 We have studied the properties of matrices and found out that there are more ways that we can solve Linear Systems. In Section 7.3, we learned that we can use
Lecture 2 Systems of Linear Equations and Matrices, Continued
Lecture 2 Systems of Linear Equations and Matrices, Continued Math 19620 Outline of Lecture Algorithm for putting a matrix in row reduced echelon form - i.e. Gauss-Jordan Elimination Number of Solutions
7.6 The Inverse of a Square Matrix
7.6 The Inverse of a Square Matrix Copyright Cengage Learning. All rights reserved. What You Should Learn Verify that two matrices are inverses of each other. Use Gauss-Jordan elimination to find inverses
Section 6.2 Larger Systems of Linear Equations
Section 6.2 Larger Systems of Linear Equations Gaussian Elimination In general, to solve a system of linear equations using its augmented matrix, we use elementary row operations to arrive at a matrix
Gauss-Jordan Row Reduction and Reduced Row Echelon Form
Gauss-Jordan Row Reduction and Reduced Row Echelon Form If we put the augmented matrix of a linear system in reduced row-echelon form, then we don t need to back-substitute to solve the system. To put
Elementary matrices, continued. To summarize, we have identified 3 types of row operations and their corresponding
Elementary matrices, continued To summarize, we have identified 3 types of row operations and their corresponding elementary matrices. If you check the previous examples, you ll find that these matrices
Chapter 1: Systems of Linear Equations and Matrices
: Systems of Linear Equations and Matrices Multiple Choice Questions. Which of the following equations is linear? (A) x + 3x 3 + 4x 4 3 = 5 (B) 3x x + x 3 = 5 (C) 5x + 5 x x 3 = x + cos (x ) + 4x 3 = 7.
The matrix will only be consistent if the last entry of row three is 0, meaning 2b 3 + b 2 b 1 = 0.
) Find all solutions of the linear system. Express the answer in vector form. x + 2x + x + x 5 = 2 2x 2 + 2x + 2x + x 5 = 8 x + 2x + x + 9x 5 = 2 2 Solution: Reduce the augmented matrix [ 2 2 2 8 ] to
Math 2331 Linear Algebra
1.1 Linear System Math 2331 Linear Algebra 1.1 Systems of Linear Equations Shang-Huan Chiu Department of Mathematics, University of Houston schiu@math.uh.edu math.uh.edu/ schiu/ Shang-Huan Chiu, University
Chapter 3. Linear Equations. Josef Leydold Mathematical Methods WS 2018/19 3 Linear Equations 1 / 33
Chapter 3 Linear Equations Josef Leydold Mathematical Methods WS 2018/19 3 Linear Equations 1 / 33 Lineares Gleichungssystem System of m linear equations in n unknowns: a 11 x 1 + a 12 x 2 + + a 1n x n
Linear System Equations
King Saud University September 24, 2018 Table of contents 1 2 3 4 Definition A linear system of equations with m equations and n unknowns is defined as follows: a 1,1 x 1 + a 1,2 x 2 + + a 1,n x n = b
is a 3 4 matrix. It has 3 rows and 4 columns. The first row is the horizontal row [ ]
![is a 3 4 matrix. It has 3 rows and 4 columns. The first row is the horizontal row [ ]](/thumbs/82/86615780.jpg "is a 3 4 matrix. It has 3 rows and 4 columns. The first row is the horizontal row [ ]")
Matrices: Definition: An m n matrix, A m n is a rectangular array of numbers with m rows and n columns: a, a, a,n a, a, a,n A m,n =...... a m, a m, a m,n Each a i,j is the entry at the i th row, j th column.
3. Replace any row by the sum of that row and a constant multiple of any other row.
Section. Solution of Linear Systems by Gauss-Jordan Method A matrix is an ordered rectangular array of numbers, letters, symbols or algebraic expressions. A matrix with m rows and n columns has size or
Elementary Linear Algebra
Elementary Linear Algebra Linear algebra is the study of; linear sets of equations and their transformation properties. Linear algebra allows the analysis of; rotations in space, least squares fitting,
Solving Systems of Linear Equations Using Matrices
Solving Systems of Linear Equations Using Matrices What is a Matrix? A matrix is a compact grid or array of numbers. It can be created from a system of equations and used to solve the system of equations.
4 Elementary matrices, continued
4 Elementary matrices, continued We have identified 3 types of row operations and their corresponding elementary matrices. If you check the previous examples, you ll find that these matrices are constructed
4 Elementary matrices, continued
4 Elementary matrices, continued We have identified 3 types of row operations and their corresponding elementary matrices. To repeat the recipe: These matrices are constructed by performing the given row
Extra Problems: Chapter 1
MA131 (Section 750002): Prepared by Asst.Prof.Dr.Archara Pacheenburawana 1 Extra Problems: Chapter 1 1. In each of the following answer true if the statement is always true and false otherwise in the space
Chapter 6. Linear Independence. Chapter 6
Linear Independence Linear Dependence/Independence A set of vectors {v, v 2,..., v p } is linearly dependent if we can express the zero vector, 0, as a non-trivial linear combination of the vectors. α
Math 2030 Assignment 5 Solutions
Math 030 Assignment 5 Solutions Question 1: Which of the following sets of vectors are linearly independent? If the set is linear dependent, find a linear dependence relation for the vectors (a) {(1, 0,
MATH 54 - WORKSHEET 1 MONDAY 6/22
MATH 54 - WORKSHEET 1 MONDAY 6/22 Row Operations: (1 (Replacement Add a multiple of one row to another row. (2 (Interchange Swap two rows. (3 (Scaling Multiply an entire row by a nonzero constant. A matrix
6-2 Matrix Multiplication, Inverses and Determinants
Find AB and BA, if possible. 1. A = A = ; A is a 1 2 matrix and B is a 2 2 matrix. Because the number of columns of A is equal to the number of rows of B, AB exists. To find the first entry of AB, find
CHAPTER 8: Matrices and Determinants
(Exercises for Chapter 8: Matrices and Determinants) E.8.1 CHAPTER 8: Matrices and Determinants (A) means refer to Part A, (B) means refer to Part B, etc. Most of these exercises can be done without a
Chapter 2 Notes, Linear Algebra 5e Lay
Contents.1 Operations with Matrices..................................1.1 Addition and Subtraction.............................1. Multiplication by a scalar............................ 3.1.3 Multiplication
EXAM. Exam #1. Math 2360, Second Summer Session, April 24, 2001 ANSWERS
i i EXAM Exam #1 Math 2360, Second Summer Session, 2002 April 24, 2001 ANSWERS i 50 pts. Problem 1. In each part you are given the augmented matrix of a system of linear equations, with the coefficent
Math 1314 Week #14 Notes
Math 3 Week # Notes Section 5.: A system of equations consists of two or more equations. A solution to a system of equations is a point that satisfies all the equations in the system. In this chapter,
Finite Mathematics. Linear Equations (Chapter 4) Optimization (Chapter 5) Finite Probability (Chapter 6) Markov Chains (Chapter 9) Finite Mathematics
Finite Mathematics Linear Equations (Chapter ) Optimization (Chapter ) Finite Probability (Chapter 6) Markov Chains (Chapter 9) Finite Mathematics Linear Algebra Probability Linear Equations Simple Method
Lecture 1 Systems of Linear Equations and Matrices
Lecture 1 Systems of Linear Equations and Matrices Math 19620 Outline of Course Linear Equations and Matrices Linear Transformations, Inverses Bases, Linear Independence, Subspaces Abstract Vector Spaces
Solving Linear Systems Using Gaussian Elimination
Solving Linear Systems Using Gaussian Elimination DEFINITION: A linear equation in the variables x 1,..., x n is an equation that can be written in the form a 1 x 1 +...+a n x n = b, where a 1,...,a n
Problem Sheet 1 with Solutions GRA 6035 Mathematics
Problem Sheet 1 with Solutions GRA 6035 Mathematics BI Norwegian Business School 2 Problems 1. From linear system to augmented matrix Write down the coefficient matrix and the augmented matrix of the following
Linear Algebra I Lecture 10
Linear Algebra I Lecture 10 Xi Chen 1 1 University of Alberta January 30, 2019 Outline 1 Gauss-Jordan Algorithm ] Let A = [a ij m n be an m n matrix. To reduce A to a reduced row echelon form using elementary
Number of solutions of a system
Roberto s Notes on Linear Algebra Chapter 3: Linear systems and matrices Section 7 Number of solutions of a system What you need to know already: How to solve a linear system by using Gauss- Jordan elimination.
Chapter 1: Systems of linear equations and matrices. Section 1.1: Introduction to systems of linear equations
Chapter 1: Systems of linear equations and matrices Section 1.1: Introduction to systems of linear equations Definition: A linear equation in n variables can be expressed in the form a 1 x 1 + a 2 x 2
Chapter 1. Vectors, Matrices, and Linear Spaces
1.4 Solving Systems of Linear Equations 1 Chapter 1. Vectors, Matrices, and Linear Spaces 1.4. Solving Systems of Linear Equations Note. We give an algorithm for solving a system of linear equations (called
9.1 - Systems of Linear Equations: Two Variables
9.1 - Systems of Linear Equations: Two Variables Recall that a system of equations consists of two or more equations each with two or more variables. A solution to a system in two variables is an ordered
7.5 Operations with Matrices. Copyright Cengage Learning. All rights reserved.
7.5 Operations with Matrices Copyright Cengage Learning. All rights reserved. What You Should Learn Decide whether two matrices are equal. Add and subtract matrices and multiply matrices by scalars. Multiply
Linear Equations in Linear Algebra
1 Linear Equations in Linear Algebra 1.1 SYSTEMS OF LINEAR EQUATIONS LINEAR EQUATION,, 1 n A linear equation in the variables equation that can be written in the form a a a b 1 1 2 2 n n a a is an where
Chapter 2. Systems of Equations and Augmented Matrices. Creighton University
Chapter Section - Systems of Equations and Augmented Matrices D.S. Malik Creighton University Systems of Linear Equations Common ways to solve a system of equations: Eliminationi Substitution Elimination
1 - Systems of Linear Equations
1 - Systems of Linear Equations 1.1 Introduction to Systems of Linear Equations Almost every problem in linear algebra will involve solving a system of equations. ü LINEAR EQUATIONS IN n VARIABLES We are
Exam 3 Review Math 118 Sections 1 and 2
Exam 3 Review Math 118 Sections 1 and 2 This exam will cover sections 5.3-5.6, 6.1-6.3 and 7.1-7.3 of the textbook. No books, notes, calculators or other aids are allowed on this exam. There is no time
8-15. Stop by or call (630)
To review the basics Matrices, what they represent, and how to find sum, scalar product, product, inverse, and determinant of matrices, watch the following set of YouTube videos. They are followed by several
1. (3pts) State three of the properties of matrix multiplication.
Math 125 Exam 2 Version 1 October 23, 2006 60 points possible 1. (a) (3pts) State three of the properties of matrix multiplication. Solution: From page 72 of the notes: Theorem: The Properties of Matrix
Midterm 1 Review. Written by Victoria Kala SH 6432u Office Hours: R 12:30 1:30 pm Last updated 10/10/2015
Midterm 1 Review Written by Victoria Kala vtkala@math.ucsb.edu SH 6432u Office Hours: R 12:30 1:30 pm Last updated 10/10/2015 Summary This Midterm Review contains notes on sections 1.1 1.5 and 1.7 in your
Linear Algebra I Lecture 8
Linear Algebra I Lecture 8 Xi Chen 1 1 University of Alberta January 25, 2019 Outline 1 2 Gauss-Jordan Elimination Given a system of linear equations f 1 (x 1, x 2,..., x n ) = 0 f 2 (x 1, x 2,..., x n
MAC1105-College Algebra. Chapter 5-Systems of Equations & Matrices
MAC05-College Algebra Chapter 5-Systems of Equations & Matrices 5. Systems of Equations in Two Variables Solving Systems of Two Linear Equations/ Two-Variable Linear Equations A system of equations is
Matrices and Determinants
Math Assignment Eperts is a leading provider of online Math help. Our eperts have prepared sample assignments to demonstrate the quality of solution we provide. If you are looking for mathematics help
22A-2 SUMMER 2014 LECTURE 5
A- SUMMER 0 LECTURE 5 NATHANIEL GALLUP Agenda Elimination to the identity matrix Inverse matrices LU factorization Elimination to the identity matrix Previously, we have used elimination to get a system
Least squares problems Linear Algebra with Computer Science Application
Linear Algebra with Computer Science Application April 8, 018 1 Least Squares Problems 11 Least Squares Problems What do you do when Ax = b has no solution? Inconsistent systems arise often in applications
Chapter 4. Solving Systems of Equations. Chapter 4
Solving Systems of Equations 3 Scenarios for Solutions There are three general situations we may find ourselves in when attempting to solve systems of equations: 1 The system could have one unique solution.
Math "Matrix Approach to Solving Systems" Bibiana Lopez. November Crafton Hills College. (CHC) 6.3 November / 25
Math 102 6.3 "Matrix Approach to Solving Systems" Bibiana Lopez Crafton Hills College November 2010 (CHC) 6.3 November 2010 1 / 25 Objectives: * Define a matrix and determine its order. * Write the augmented
Finite Mathematics Chapter 2. where a, b, c, d, h, and k are real numbers and neither a and b nor c and d are both zero.
Finite Mathematics Chapter 2 Section 2.1 Systems of Linear Equations: An Introduction Systems of Equations Recall that a system of two linear equations in two variables may be written in the general form
MATH 152 Exam 1-Solutions 135 pts. Write your answers on separate paper. You do not need to copy the questions. Show your work!!!
MATH Exam -Solutions pts Write your answers on separate paper. You do not need to copy the questions. Show your work!!!. ( pts) Find the reduced row echelon form of the matrix Solution : 4 4 6 4 4 R R
Pre-Calculus I. For example, the system. x y 2 z. may be represented by the augmented matrix
Pre-Calculus I 8.1 Matrix Solutions to Linear Systems A matrix is a rectangular array of elements. o An array is a systematic arrangement of numbers or symbols in rows and columns. Matrices (the plural
Exam. Name. Domain: (0, ) Range: (-, ) Domain: (0, ) Range: (-, ) Domain: (-, ) Range: (0, ) Domain: (-, ) Range: (0, ) y
Eam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Graph the function and write the domain and range in interval notation. ) f () = 5 B) 0 0
System of Linear Equations. Slide for MA1203 Business Mathematics II Week 1 & 2
System of Linear Equations Slide for MA1203 Business Mathematics II Week 1 & 2 Function A manufacturer would like to know how his company s profit is related to its production level. How does one quantity
Math 1070 Exam 2. March 16, 2006
Math 1070 Exam 2 Name: March 16, 2006 1. Solve each of the following systems of equations using the indicated method: (a) Solve by graphing (include a sketch of your graph, indicating the graphing window,
Definition of Equality of Matrices. Example 1: Equality of Matrices. Consider the four matrices
IT 131: Mathematics for Science Lecture Notes 3 Source: Larson, Edwards, Falvo (2009): Elementary Linear Algebra, Sixth Edition. Matrices 2.1 Operations with Matrices This section and the next introduce
a. Define your variables. b. Construct and fill in a table. c. State the Linear Programming Problem. Do Not Solve.
Math Section. Example : The officers of a high school senior class are planning to rent buses and vans for a class trip. Each bus can transport 4 students, requires chaperones, and costs $, to rent. Each
Math 343 Midterm I Fall 2006 sections 002 and 003 Instructor: Scott Glasgow
Math 343 Midterm I Fall 006 sections 00 003 Instructor: Scott Glasgow 1 Assuming A B are invertible matrices of the same size, prove that ( ) 1 1 AB B A 1 = (11) B A 1 1 is the inverse of AB if only if
Solving Ax = b w/ different b s: LU-Factorization
Solving Ax = b w/ different b s: LU-Factorization Linear Algebra Josh Engwer TTU 14 September 2015 Josh Engwer (TTU) Solving Ax = b w/ different b s: LU-Factorization 14 September 2015 1 / 21 Elementary
MTH5112 Linear Algebra I MTH5212 Applied Linear Algebra (2017/2018)
MTH5112 Linear Algebra I MTH5212 Applied Linear Algebra (2017/2018) COURSEWORK 3 SOLUTIONS Exercise ( ) 1. (a) Write A = (a ij ) n n and B = (b ij ) n n. Since A and B are diagonal, we have a ij = 0 and
MIDTERM 1 - SOLUTIONS
MIDTERM - SOLUTIONS MATH 254 - SUMMER 2002 - KUNIYUKI CHAPTERS, 2, GRADED OUT OF 75 POINTS 2 50 POINTS TOTAL ) Use either Gaussian elimination with back-substitution or Gauss-Jordan elimination to solve
Math 1070 Exam 2. March 16, 2006
Math 1070 Exam 2 Name: March 16, 2006 1. Solve each of the following systems of equations using the indicated method: (a) Solve by graphing (include a sketch of your graph, indicating the graphing window,
Section 1.1: Systems of Linear Equations
Section 1.1: Systems of Linear Equations Two Linear Equations in Two Unknowns Recall that the equation of a line in 2D can be written in standard form: a 1 x 1 + a 2 x 2 = b. Definition. A 2 2 system of
Check that your exam contains 20 multiple-choice questions, numbered sequentially.
MATH 22 MAKEUP EXAMINATION Fall 26 VERSION A NAME STUDENT NUMBER INSTRUCTOR SECTION NUMBER On your scantron, write and bubble your PSU ID, Section Number, and Test Version. Failure to correctly code these
Lecture 12: Solving Systems of Linear Equations by Gaussian Elimination
Lecture 12: Solving Systems of Linear Equations by Gaussian Elimination Winfried Just, Ohio University September 22, 2017 Review: The coefficient matrix Consider a system of m linear equations in n variables.
1. Determine by inspection which of the following sets of vectors is linearly independent. 3 3.
1. Determine by inspection which of the following sets of vectors is linearly independent. (a) (d) 1, 3 4, 1 { [ [,, 1 1] 3]} (b) 1, 4 5, (c) 3 6 (e) 1, 3, 4 4 3 1 4 Solution. The answer is (a): v 1 is
Chapter If M is the matrix 0 0 1, then M 100 is. (c) (a) (b) (d) (e) None of the above. (b) 2.
Chapter 2. If M is the matrix, then M is (a) (b) (c) (d) (e) None of the above 3 6 2. If A is a 3 3 matrix such that A = and A 4 =, then the product A 7 is 2 5 8 (a) (b) 2 (c) 9 (d) (e) Not uniquely determined
Chapter 9: Systems of Equations and Inequalities
Chapter 9: Systems of Equations and Inequalities 9. Systems of Equations Solve the system of equations below. By this we mean, find pair(s) of numbers (x, y) (if possible) that satisfy both equations.
Systems of Linear Equations in Two Variables. Break Even. Example. 240x x This is when total cost equals total revenue.
Systems of Linear Equations in Two Variables 1 Break Even This is when total cost equals total revenue C(x) = R(x) A company breaks even when the profit is zero P(x) = R(x) C(x) = 0 2 R x 565x C x 6000
Matrix equation Ax = b
Fall 2017 Matrix equation Ax = b Authors: Alexander Knop Institute: UC San Diego Previously On Math 18 DEFINITION If v 1,..., v l R n, then a set of all linear combinations of them is called Span {v 1,...,
GAUSSIAN ELIMINATION AND LU DECOMPOSITION (SUPPLEMENT FOR MA511)
GAUSSIAN ELIMINATION AND LU DECOMPOSITION (SUPPLEMENT FOR MA511) D. ARAPURA Gaussian elimination is the go to method for all basic linear classes including this one. We go summarize the main ideas. 1.
Multiple Choice Questions
Multiple Choice Questions There is no penalty for guessing. Three points per question, so a total of 48 points for this section.. What is the complete relationship between homogeneous linear systems of
System of Linear Equations
Chapter 7 - S&B Gaussian and Gauss-Jordan Elimination We will study systems of linear equations by describing techniques for solving such systems. The preferred solution technique- Gaussian elimination-
MathQuest: Linear Algebra. 1. Which of the following operations on an augmented matrix could change the solution set of a system?
MathQuest: Linear Algebra Gaussian Elimination 1. Which of the following operations on an augmented matrix could change the solution set of a system? (a) Interchanging two rows (b) Multiplying one row
6-1 Study Guide and Intervention Multivariable Linear Systems and Row Operations
6-1 Study Guide and Intervention Multivariable Linear Systems and Row Operations Gaussian Elimination You can solve a system of linear equations using matrices. Solving a system by transforming it into
Systems of equation and matrices
Systems of equation and matrices Jean-Luc Bouchot jean-luc.bouchot@drexel.edu February 23, 2013 Warning This is a work in progress. I can not ensure it to be mistake free at the moment. It is also lacking
Linear Equations in Linear Algebra
1 Linear Equations in Linear Algebra 1.1 SYSTEMS OF LINEAR EQUATIONS LINEAR EQUATION x 1,, x n A linear equation in the variables equation that can be written in the form a 1 x 1 + a 2 x 2 + + a n x n
b for the linear system x 1 + x 2 + a 2 x 3 = a x 1 + x 3 = 3 x 1 + x 2 + 9x 3 = 3 ] 1 1 a 2 a
![b for the linear system x 1 + x 2 + a 2 x 3 = a x 1 + x 3 = 3 x 1 + x 2 + 9x 3 = 3 ] 1 1 a 2 a](/thumbs/71/65559666.jpg "b for the linear system x 1 + x 2 + a 2 x 3 = a x 1 + x 3 = 3 x 1 + x 2 + 9x 3 = 3 ] 1 1 a 2 a")
Practice Exercises for Exam Exam will be on Monday, September 8, 7. The syllabus for Exam consists of Sections One.I, One.III, Two.I, and Two.II. You should know the main definitions, results and computational
Section Gaussian Elimination
Section. - Gaussian Elimination A matrix is said to be in row echelon form (REF) if it has the following properties:. The first nonzero entry in any row is a. We call this a leading one or pivot one..
March 19 - Solving Linear Systems
March 19 - Solving Linear Systems Welcome to linear algebra! Linear algebra is the study of vectors, vector spaces, and maps between vector spaces. It has applications across data analysis, computer graphics,
ANSWERS. Answer: Perform combo(3,2,-1) on I then combo(1,3,-4) on the result. The elimination matrix is
MATH 227-2 Sample Exam 1 Spring 216 ANSWERS 1. (1 points) (a) Give a counter example or explain why it is true. If A and B are n n invertible, and C T denotes the transpose of a matrix C, then (AB 1 )
Systems of Linear Equations. By: Tri Atmojo Kusmayadi and Mardiyana Mathematics Education Sebelas Maret University
Systems of Linear Equations By: Tri Atmojo Kusmayadi and Mardiyana Mathematics Education Sebelas Maret University Standard of Competency: Understanding the properties of systems of linear equations, matrices,
Math Week in Review #3 - Exam 1 Review
Math 166 Exam 1 Review Fall 2006 c Heather Ramsey Page 1 Math 166 - Week in Review #3 - Exam 1 Review NOTE: For reviews of the other sections on Exam 1, refer to the first page of WIR #1 and #2. Section
Section 5.3 Systems of Linear Equations: Determinants
Section 5. Systems of Linear Equations: Determinants In this section, we will explore another technique for solving systems called Cramer's Rule. Cramer's rule can only be used if the number of equations
1 Linear systems, existence, uniqueness
Jor-el Briones / Math 2F, 25 Summer Session, Practice Midterm Page of 9 Linear systems, existence, uniqueness For each part, construct an augmented matrix for a linear system with the given properties,
Exercise Sketch these lines and find their intersection.
These are brief notes for the lecture on Friday August 21, 2009: they are not complete, but they are a guide to what I want to say today. They are not guaranteed to be correct. 1. Solving systems of linear
1. Let m 1 and n 1 be two natural numbers such that m > n. Which of the following is/are true?
. Let m and n be two natural numbers such that m > n. Which of the following is/are true? (i) A linear system of m equations in n variables is always consistent. (ii) A linear system of n equations in
10.3 Matrices and Systems Of
10.3 Matrices and Systems Of Linear Equations Copyright Cengage Learning. All rights reserved. Objectives Matrices The Augmented Matrix of a Linear System Elementary Row Operations Gaussian Elimination
Row Reduced Echelon Form
Math 40 Row Reduced Echelon Form Solving systems of linear equations lies at the heart of linear algebra. In high school we learn to solve systems in or variables using elimination and substitution of
Math Studio College Algebra
Math 100 - Studio College Algebra Rekha Natarajan Kansas State University November 19, 2014 Systems of Equations Systems of Equations A system of equations consists of Systems of Equations A system of
Matrices. A matrix is a method of writing a set of numbers using rows and columns. Cells in a matrix can be referenced in the form.
Matrices A matrix is a method of writing a set of numbers using rows and columns. 1 2 3 4 3 2 1 5 7 2 5 4 2 0 5 10 12 8 4 9 25 30 1 1 Reading Information from a Matrix Cells in a matrix can be referenced
Recall, we solved the system below in a previous section. Here, we learn another method. x + 4y = 14 5x + 3y = 2
We will learn how to use a matrix to solve a system of equations. College algebra Class notes Matrices and Systems of Equations (section 6.) Recall, we solved the system below in a previous section. Here,
MAC Module 1 Systems of Linear Equations and Matrices I
MAC 2103 Module 1 Systems of Linear Equations and Matrices I 1 Learning Objectives Upon completing this module, you should be able to: 1. Represent a system of linear equations as an augmented matrix.