# Section Gaussian Elimination

Size: px
Start display at page:

Download "Section Gaussian Elimination"

## Transcription

1 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.. The leading one in any successive row must be to the right of the leading one in the previous row. This gives the matrix an "echelon" form.. Any rows consisting entirely of zeros must be grouped together at the bottom of the matrix. A matrix is said to be in reduced row echelon form (RREF) if it is in REF and each column that contains a leading one has zeros everywhere else in that column. Determine whether matrices are in REF only, RREF, or not REF The location of leading ones in any REF for a given matrix will always be located in the same position. For augmented matrices, variables that correspond to the leading ones in any REF are called leading variables or fixed variables. The remaining variables are called free variables. When we have free variables for a consistent system of linear equations, then that system will have infinitely many solutions. In this case, we will express the solution using a set of parametric equations. Here is a REF for the augmented matrix of a system of linear equations. State which variables are fixed and which are free

2 In each case, a REF for the augmented matrix of a system of linear equations is given. Give the number of equations, the number of unknowns, and the solution (if any). Parameterize solution as needed. # of equations # unknowns Solution # of equations # unknowns Solution # of equations # unknowns Solution # of equations # unknowns Solution

3 In each case, write out a possible RREF of the augmented matrix for the system described: 5. Three equations and unknowns with solution x 4, x, x Three equations and unknowns with solution x t, x 5t, x t. 7. Three equations and unknowns with no solution. 8. Four equations and unknowns with solution x, x. 9. A homogeneous system of equations in unknowns that has infinitely many solutions.

4 We will now show how to get any augmented matrix into a REF using a step-by-step elimination method that uses only elementary row operations. Recall the elementary row operations:. swap any two rows.. multiply row by a nonzero constant.. Add a multiple of one row to another. The method we will show that reduces an augmented matrix to a REF is called Gaussian elimination (in honor of the great German mathematician Carl Friedrich Gauss). This procedure (or algorithm) consists of what is referred to as the forward phase in which zeros are introduced below the leading ones. If we continue this algorithm and add the backward phase (in which zeros are introduced above the leading ones), we can obtain the RREF of the augmented matrix. The procedure that uses both forward and backward phases to put the augmented matrix into its RREF is called Gauss-Jordan elimination (to also honor the German engineer Wilhelm Jordan). When solving with Gaussian elimination, the solution is obtained by writing out the system of equations obtained from the REF of augmented matrix and then using back substitution. Here is the idea of the algorithm: Step : Get a leading one in the upper right corner by either interchanging rows if this entry is zero and/or multiplying the row by the reciprocal of this entry. Step : Get zeros in the rows beneath this leading (or pivot) one by adding appropriate multiples of first row to the row with the entry you want to zero out. The multiple is the opposite of the entry you want to zero out. 0 0 Step : Shift down to the second row second column. If there is a zero here, look to see if any entry beneath it is nonzero. If not move to the next column in the second row and repeat this step. If there is a nonzero entry, interchange rows to get the nonzero entry in the second row. Multiply the second row by the reciprocal of this nonzero entry to get another leading one. 0 0 Step 4: Get zeros in the rows beneath this leading (or pivot) one by adding appropriate multiples of first row to the row with the entry you want to zero out. The multiple is the opposite of the entry you want to zero out Continue this process until you have it in a REF. By adding multiples of rows and using the pivot ones, you can now continue to the RREF. 4

5 Solve the system shown to the left below using Gaussian Elimination with back substitution & then with Gauss-Jordan Elimination. x x x 5 5x x x 0 x x x 6x 4x x The system of equations yields the augmented matrix shown above to the right of the system. We first will show Gaussian Elimination with back substitution. We proceed by using the elementary row operations to get a REF. R R R R 5R R R R R 4 R 4 6R R R R R R R R 7 The resulting system from this is and we have x x x 5 x x 75 x 7 This is now in REF form x 5 x x x 75 x so that by back substitution we have x 7 x 75 7 x

6 We could have also obtained solution directly had we solved using the Gauss-Jordan elimination. This requires transforming the REF form above into RREF R R R 0 75 R R 7R R R R This is now in RREF form The corresponding system to the RREF gives the same solution directly. x 7 x x 4 6

7 Solve the following system using Gauss-Jordan elimination 5x 0x x x 4 5x 5 0 x 4x x 4 7x 5 7 x x x 5 7

8 Recall that a homogeneous system is one in which all the constant terms are zero. That is, if the system is of the form a x a x a x a n x n 0 a x a x a x a n x n 0 a x a x a x a n x n 0 a m x a m x a m x a mn x n 0 Every homogeneous system is consistent because all such systems have x 0, x 0,, x n 0 as a solution. This solution is called the trivial solution. If there are other solutions to the system than they are called nontrivial solutions. Examples of homogeneous systems: x 4x x 0 5x 0x 4x x 4 0 x 5x x 0 6x 9x x 4 0 6x x 7x 0 7x x 8x 0 Since every homogeneous system is consistent, there are only two possibilities for its solution:. The system has only the trivial solution.. The system has infinitely many solutions in addition to the trivial solution. When performing row operations on matrices, columns of zeros are not altered. Thus, any REF of a homogeneous system will have a last column of all zeros. Theorem: If a homogeneous linear system has n unknowns (variables), and if the RREF of its augmented matrix has r nonzero rows (i.e. rows with a leading one), then the system has n r free variables. Theorem: A homogeneous linear system with fewer equations than unknowns has infinitely many solutions. Note that this last theorem does NOT apply to nonhomogeneous linear systems as such a system may not be consistent. However, it can be proved that a nonhomogeneous linear systems with fewer equations than unknowns that is consistent has infinitely many solutions. 8

### 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

More information

### 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.

More information

### 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

More information

### 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

More information

### Lectures on Linear Algebra for IT

Lectures on Linear Algebra for IT by Mgr. Tereza Kovářová, Ph.D. following content of lectures by Ing. Petr Beremlijski, Ph.D. Department of Applied Mathematics, VSB - TU Ostrava Czech Republic 2. Systems

More information

### 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

More information

### 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-

More information

### 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

More information

### Row Reduction and Echelon Forms

Row Reduction and Echelon Forms 1 / 29 Key Concepts row echelon form, reduced row echelon form pivot position, pivot, pivot column basic variable, free variable general solution, parametric solution existence

More information

### DM559 Linear and Integer Programming. Lecture 2 Systems of Linear Equations. Marco Chiarandini

DM559 Linear and Integer Programming Lecture Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark Outline 1. Outline 1. 3 A Motivating Example You are organizing

More information

### 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

More information

### Notes on Row Reduction

Notes on Row Reduction Francis J. Narcowich Department of Mathematics Texas A&M University September The Row-Reduction Algorithm The row-reduced form of a matrix contains a great deal of information, both

More information

### Determine whether the following system has a trivial solution or non-trivial solution:

Practice Questions Lecture # 7 and 8 Question # Determine whether the following system has a trivial solution or non-trivial solution: x x + x x x x x The coefficient matrix is / R, R R R+ R The corresponding

More information

### 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,

More information

### Matrices and RRE Form

Matrices and RRE Form Notation R is the real numbers, C is the complex numbers (we will only consider complex numbers towards the end of the course) is read as an element of For instance, x R means that

More information

### Lecture 3: Gaussian Elimination, continued. Lecture 3: Gaussian Elimination, continued

Definition The process of solving a system of linear equations by converting the system to an augmented matrix is called Gaussian Elimination. The general strategy is as follows: Convert the system of

More information

### Lecture 4: Gaussian Elimination and Homogeneous Equations

Lecture 4: Gaussian Elimination and Homogeneous Equations Reduced Row Echelon Form An augmented matrix associated to a system of linear equations is said to be in Reduced Row Echelon Form (RREF) if the

More information

### 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

More information

### 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

More information

### 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

More information

### 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

More information

### Example: 2x y + 3z = 1 5y 6z = 0 x + 4z = 7. Definition: Elementary Row Operations. Example: Type I swap rows 1 and 3

Math 0 Row Reduced Echelon Form Techniques for solving systems of linear equations lie at the heart of linear algebra. In high school we learn to solve systems with or variables using elimination and substitution

More information

### 1300 Linear Algebra and Vector Geometry Week 2: Jan , Gauss-Jordan, homogeneous matrices, intro matrix arithmetic

1300 Linear Algebra and Vector Geometry Week 2: Jan 14 18 1.2, 1.3... Gauss-Jordan, homogeneous matrices, intro matrix arithmetic R. Craigen Office: MH 523 Email: craigenr@umanitoba.ca Winter 2019 What

More information

### 1 Last time: linear systems and row operations

1 Last time: linear systems and row operations Here s what we did last time: a system of linear equations or linear system is a list of equations a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22

More information

### 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

More information

### CHAPTER 9: Systems of Equations and Matrices

MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 9: Systems of Equations and Matrices 9.1 Systems of Equations in Two Variables 9.2 Systems of Equations in Three Variables

More information

### Chapter 1: Systems of Linear Equations

Chapter : Systems of Linear Equations February, 9 Systems of linear equations Linear systems Lecture A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where

More information

### Review for Exam Find all a for which the following linear system has no solutions, one solution, and infinitely many solutions.

Review for Exam. Find all a for which the following linear system has no solutions, one solution, and infinitely many solutions. x + y z = 2 x + 2y + z = 3 x + y + (a 2 5)z = a 2 The augmented matrix for

More information

### Relationships Between Planes

Relationships Between Planes Definition: consistent (system of equations) A system of equations is consistent if there exists one (or more than one) solution that satisfies the system. System 1: {, System

More information

### Solving Systems of Linear Equations

LECTURE 5 Solving Systems of Linear Equations Recall that we introduced the notion of matrices as a way of standardizing the expression of systems of linear equations In today s lecture I shall show how

More information

### 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,

More information

### Chapter 5. Linear Algebra. A linear (algebraic) equation in. unknowns, x 1, x 2,..., x n, is. an equation of the form

Chapter 5. Linear Algebra A linear (algebraic) equation in n unknowns, x 1, x 2,..., x n, is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b where a 1, a 2,..., a n and b are real numbers. 1

More information

### 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

More information

### Chapter 5. Linear Algebra. Sections A linear (algebraic) equation in. unknowns, x 1, x 2,..., x n, is. an equation of the form

Chapter 5. Linear Algebra Sections 5.1 5.3 A linear (algebraic) equation in n unknowns, x 1, x 2,..., x n, is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b where a 1, a 2,..., a n and b are

More information

### 1300 Linear Algebra and Vector Geometry

1300 Linear Algebra and Vector Geometry R. Craigen Office: MH 523 Email: craigenr@umanitoba.ca May-June 2017 Introduction: linear equations Read 1.1 (in the text that is!) Go to course, class webpages.

More information

### SOLVING Ax = b: GAUSS-JORDAN ELIMINATION [LARSON 1.2]

SOLVING Ax = b: GAUSS-JORDAN ELIMINATION [LARSON.2 EQUIVALENT LINEAR SYSTEMS: Two m n linear systems are equivalent both systems have the exact same solution sets. When solving a linear system Ax = b,

More information

### 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

More information

### 3.4 Elementary Matrices and Matrix Inverse

Math 220: Summer 2015 3.4 Elementary Matrices and Matrix Inverse A n n elementary matrix is a matrix which is obtained from the n n identity matrix I n n by a single elementary row operation. Elementary

More information

### CHAPTER 9: Systems of Equations and Matrices

MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 9: Systems of Equations and Matrices 9.1 Systems of Equations in Two Variables 9.2 Systems of Equations in Three Variables

More information

### Chapter 5. Linear Algebra. Sections A linear (algebraic) equation in. unknowns, x 1, x 2,..., x n, is. an equation of the form

Chapter 5. Linear Algebra Sections 5.1 5.3 A linear (algebraic) equation in n unknowns, x 1, x 2,..., x n, is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b where a 1, a 2,..., a n and b are

More information

### Linear equations in linear algebra

Linear equations in linear algebra Samy Tindel Purdue University Differential equations and linear algebra - MA 262 Taken from Differential equations and linear algebra Pearson Collections Samy T. Linear

More information

### 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.

More information

### 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

More information

### 5.7 Cramer's Rule 1. Using Determinants to Solve Systems Assumes the system of two equations in two unknowns

5.7 Cramer's Rule 1. Using Determinants to Solve Systems Assumes the system of two equations in two unknowns (1) possesses the solution and provided that.. The numerators and denominators are recognized

More information

### Matrices and systems of linear equations

Matrices and systems of linear equations Samy Tindel Purdue University Differential equations and linear algebra - MA 262 Taken from Differential equations and linear algebra by Goode and Annin Samy T.

More information

### INVERSE OF A MATRIX [2.2]

INVERSE OF A MATRIX [2.2] The inverse of a matrix: Introduction We have a mapping from R n to R n represented by a matrix A. Can we invert this mapping? i.e. can we find a matrix (call it B for now) such

More information

### 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

More information

### 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

More information

### 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

More information

### The definition of a vector space (V, +, )

The definition of a vector space (V, +, ) 1. For any u and v in V, u + v is also in V. 2. For any u and v in V, u + v = v + u. 3. For any u, v, w in V, u + ( v + w) = ( u + v) + w. 4. There is an element

More information

### Matrices, Row Reduction of Matrices

Matrices, Row Reduction of Matrices October 9, 014 1 Row Reduction and Echelon Forms In the previous section, we saw a procedure for solving systems of equations It is simple in that it consists of only

More information

### EBG # 3 Using Gaussian Elimination (Echelon Form) Gaussian Elimination: 0s below the main diagonal

EBG # 3 Using Gaussian Elimination (Echelon Form) Gaussian Elimination: 0s below the main diagonal [ x y Augmented matrix: 1 1 17 4 2 48 (Replacement) Replace a row by the sum of itself and a multiple

More information

### 6.3. MULTIVARIABLE LINEAR SYSTEMS

6.3. MULTIVARIABLE LINEAR SYSTEMS What You Should Learn Use back-substitution to solve linear systems in row-echelon form. Use Gaussian elimination to solve systems of linear equations. Solve nonsquare

More information

### 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,

More information

### 1. Solve each linear system using Gaussian elimination or Gauss-Jordan reduction. The augmented matrix of this linear system is

Solutions to Homework Additional Problems. Solve each linear system using Gaussian elimination or Gauss-Jordan reduction. (a) x + y = 8 3x + 4y = 7 x + y = 3 The augmented matrix of this linear system

More information

### Matrix Solutions to Linear Equations

Matrix Solutions to Linear Equations Augmented matrices can be used as a simplified way of writing a system of linear equations. In an augmented matrix, a vertical line is placed inside the matrix to represent

More information

### Systems of Linear Equations and Matrices

CHAPTER Systems of Linear Equations and Matrices CHAPTER CONTENTS Introduction to Systems of Linear Equations Gaussian Elimination 3 Matrices and Matrix Operations 5 4 Inverses; Algebraic Properties of

More information

### LECTURES 4/5: SYSTEMS OF LINEAR EQUATIONS

LECTURES 4/5: SYSTEMS OF LINEAR EQUATIONS MA1111: LINEAR ALGEBRA I, MICHAELMAS 2016 1 Linear equations We now switch gears to discuss the topic of solving linear equations, and more interestingly, systems

More information

### 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

More information

### Linear Algebra Handout

Linear Algebra Handout References Some material and suggested problems are taken from Fundamentals of Matrix Algebra by Gregory Hartman, which can be found here: http://www.vmi.edu/content.aspx?id=779979.

More information

### 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

More information

### 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.

More information

### Lecture 2e Row Echelon Form (pages 73-74)

Lecture 2e Row Echelon Form (pages 73-74) At the end of Lecture 2a I said that we would develop an algorithm for solving a system of linear equations, and now that we have our matrix notation, we can proceed

More information

### 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

More information

### 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

More information

### Chapter 1 Linear Equations. 1.1 Systems of Linear Equations

Chapter Linear Equations. Systems of Linear Equations A linear equation in the n variables x, x 2,..., x n is one that can be expressed in the form a x + a 2 x 2 + + a n x n = b where a, a 2,..., a n and

More information

### 13. Systems of Linear Equations 1

13. Systems of Linear Equations 1 Systems of linear equations One of the primary goals of a first course in linear algebra is to impress upon the student how powerful matrix methods are in solving systems

More information

### 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

More information

### Section Gauss Elimination for Systems of Linear Equations

Section 4.3 - Gauss Elimination for Systems of Linear Equations What is a linear equation? What does it mean to solve a system of linear equations? What are the possible cases when solving a system of

More information

### MATH 2331 Linear Algebra. Section 1.1 Systems of Linear Equations. Finding the solution to a set of two equations in two variables: Example 1: Solve:

MATH 2331 Linear Algebra Section 1.1 Systems of Linear Equations Finding the solution to a set of two equations in two variables: Example 1: Solve: x x = 3 1 2 2x + 4x = 12 1 2 Geometric meaning: Do these

More information

### 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

More information

### Math 54 HW 4 solutions

Math 54 HW 4 solutions 2.2. Section 2.2 (a) False: Recall that performing a series of elementary row operations A is equivalent to multiplying A by a series of elementary matrices. Suppose that E,...,

More information

### Algebra & Trig. I. For example, the system. x y 2 z. may be represented by the augmented matrix

Algebra & Trig. 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

More information

### (I.D) Solving Linear Systems via Row-Reduction

(I.D) Solving Linear Systems via Row-Reduction Turning to the promised algorithmic approach to Gaussian elimination, we say an m n matrix M is in reduced-row echelon form if: the first nonzero entry of

More information

### Rectangular Systems and Echelon Forms

CHAPTER 2 Rectangular Systems and Echelon Forms 2.1 ROW ECHELON FORM AND RANK We are now ready to analyze more general linear systems consisting of m linear equations involving n unknowns a 11 x 1 + a

More information

### 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

More information

### 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

More information

### 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

More information

### Linear Equation: a 1 x 1 + a 2 x a n x n = b. x 1, x 2,..., x n : variables or unknowns

Linear Equation: a x + a 2 x 2 +... + a n x n = b. x, x 2,..., x n : variables or unknowns a, a 2,..., a n : coefficients b: constant term Examples: x + 4 2 y + (2 5)z = is linear. x 2 + y + yz = 2 is

More information

### Inverting Matrices. 1 Properties of Transpose. 2 Matrix Algebra. P. Danziger 3.2, 3.3

3., 3.3 Inverting Matrices P. Danziger 1 Properties of Transpose Transpose has higher precedence than multiplication and addition, so AB T A ( B T and A + B T A + ( B T As opposed to the bracketed expressions

More information

### 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,

More information

### REPLACE ONE ROW BY ADDING THE SCALAR MULTIPLE OF ANOTHER ROW

20 CHAPTER 1 Systems of Linear Equations REPLACE ONE ROW BY ADDING THE SCALAR MULTIPLE OF ANOTHER ROW The last type of operation is slightly more complicated. Suppose that we want to write down the elementary

More information

### PH1105 Lecture Notes on Linear Algebra.

PH05 Lecture Notes on Linear Algebra Joe Ó hógáin E-mail: johog@mathstcdie Main Text: Calculus for the Life Sciences by Bittenger, Brand and Quintanilla Other Text: Linear Algebra by Anton and Rorres Matrices

More information

### 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

More information

### x 1 2x 2 +x 3 = 0 2x 2 8x 3 = 8 4x 1 +5x 2 +9x 3 = 9

Sec 2.1 Row Operations and Gaussian Elimination Consider a system of linear equations x 1 2x 2 +x 3 = 0 2x 2 8x 3 = 8 4x 1 +5x 2 +9x 3 = 9 The coefficient matrix of the system is The augmented matrix of

More information

### 1 System of linear equations

1 System of linear equations 1.1 Two equations in two unknowns The following is a system of two linear equations in the two unknowns x and y: x y = 1 3x+4y = 6. A solution to the system is a pair (x,y)

More information

### Section 1.5. Solution Sets of Linear Systems

Section 1.5 Solution Sets of Linear Systems Plan For Today Today we will learn to describe and draw the solution set of an arbitrary system of linear equations Ax = b, using spans. Ax = b Recall: the solution

More information

### 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

More information

### Homework 1 Due: Wednesday, August 27. x + y + z = 1. x y = 3 x + y + z = c 2 2x + cz = 4

Homework 1 Due: Wednesday, August 27 1. Find all values of c for which the linear system: (a) has no solutions. (b) has exactly one solution. (c) has infinitely many solutions. (d) is consistent. x + y

More information

### Name: Section Registered In:

Name: Section Registered In: Math 125 Exam 1 Version 1 February 21, 2006 60 points possible 1. (a) (3pts) Define what it means for a linear system to be inconsistent. Solution: A linear system is inconsistent

More information

### MATH 2050 Assignment 6 Fall 2018 Due: Thursday, November 1. x + y + 2z = 2 x + y + z = c 4x + 2z = 2

MATH 5 Assignment 6 Fall 8 Due: Thursday, November [5]. For what value of c does have a solution? Is it unique? x + y + z = x + y + z = c 4x + z = Writing the system as an augmented matrix, we have c R

More information

### MODEL ANSWERS TO THE THIRD HOMEWORK

MODEL ANSWERS TO THE THIRD HOMEWORK 1 (i) We apply Gaussian elimination to A First note that the second row is a multiple of the first row So we need to swap the second and third rows 1 3 2 1 2 6 5 7 3

More information

### 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

More information

### Math 1021, Linear Algebra 1. Section: A at 10am, B at 2:30pm

Math 1021, Linear Algebra 1. Section: A at 10am, B at 2:30pm All course information is available on Moodle. Text: Nicholson, Linear algebra with applications, 7th edition. We shall cover Chapters 1,2,3,4,5:

More information

### MTH 464: Computational Linear Algebra

MTH 464: Computational Linear Algebra Lecture Outlines Exam 1 Material Dr. M. Beauregard Department of Mathematics & Statistics Stephen F. Austin State University January 9, 2018 Linear Algebra (MTH 464)

More information

### 2 Systems of Linear Equations

2 Systems of Linear Equations A system of equations of the form or is called a system of linear equations. x + 2y = 7 2x y = 4 5p 6q + r = 4 2p + 3q 5r = 7 6p q + 4r = 2 Definition. An equation involving

More information

### 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

More information

### Topics. Vectors (column matrices): Vector addition and scalar multiplication The matrix of a linear function y Ax The elements of a matrix A : A ij

Topics Vectors (column matrices): Vector addition and scalar multiplication The matrix of a linear function y Ax The elements of a matrix A : A ij or a ij lives in row i and column j Definition of a matrix

More information

### 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

More information

### 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

More information