Lecture 3 - System of Linear Equations and Matrices | Linear Algebra - Engineering Mathematics PDF Download

 Page 1


System of Linear Equations and Matrices 
 
Institute of Lifelong Learning, University of Delhi                                                pg. 1 
 
 
 
 
 
 
 
 
 
 
 
Paper: Linear Algebra 
Lesson: System of Linear Equations and Matrices 
Course Developer: Parvinder Kaur 
Department/College: Assistant Professor, Department of 
Mathematics, Motilal Nehru College, University of Delhi 
 
 
 
 
 
  
Page 2


System of Linear Equations and Matrices 
 
Institute of Lifelong Learning, University of Delhi                                                pg. 1 
 
 
 
 
 
 
 
 
 
 
 
Paper: Linear Algebra 
Lesson: System of Linear Equations and Matrices 
Course Developer: Parvinder Kaur 
Department/College: Assistant Professor, Department of 
Mathematics, Motilal Nehru College, University of Delhi 
 
 
 
 
 
  
System of Linear Equations and Matrices 
Institute of Lifelong Learning, University of Delhi                                                pg. 2 
 
Table of Contents: 
 Chapter : System of Linear Equations and Matrices 
 1. Learning outcomes 
 2. Systems of Linear Equations and Matrices 
  2.1 Introduction 
  2.2 Linear equations 
  2.3 System of Linear equations 
 3. Matrices and Elementary operations 
  3.1 Matrix representation of a linear system of equations 
  3.2 Elementary operations 
  3.3 Elementary matrices 
  3.4 Equivalent systems 
  3.5 Row equivalence  
 4. Special Matrices and their applications 
  4.1 Triangular Matrix 
  4.2 Reduced Row echelon matrix 
 4.3 Solving system of linear equations using Gaussian Elimination   
          Method 
 4.4 Solving system of linear equations using Gauss Jordan Elimination  
          Method 
 5. Rank 
  5.1 Rank of a matrix 
  5.2     Linearly Dependent and Linearly Independent Vectors 
  5.3 Row Rank and Column Rank 
  5.4 Consistency of system of linear equations 
 Exercise 
 Summary 
 References 
 
Page 3


System of Linear Equations and Matrices 
 
Institute of Lifelong Learning, University of Delhi                                                pg. 1 
 
 
 
 
 
 
 
 
 
 
 
Paper: Linear Algebra 
Lesson: System of Linear Equations and Matrices 
Course Developer: Parvinder Kaur 
Department/College: Assistant Professor, Department of 
Mathematics, Motilal Nehru College, University of Delhi 
 
 
 
 
 
  
System of Linear Equations and Matrices 
Institute of Lifelong Learning, University of Delhi                                                pg. 2 
 
Table of Contents: 
 Chapter : System of Linear Equations and Matrices 
 1. Learning outcomes 
 2. Systems of Linear Equations and Matrices 
  2.1 Introduction 
  2.2 Linear equations 
  2.3 System of Linear equations 
 3. Matrices and Elementary operations 
  3.1 Matrix representation of a linear system of equations 
  3.2 Elementary operations 
  3.3 Elementary matrices 
  3.4 Equivalent systems 
  3.5 Row equivalence  
 4. Special Matrices and their applications 
  4.1 Triangular Matrix 
  4.2 Reduced Row echelon matrix 
 4.3 Solving system of linear equations using Gaussian Elimination   
          Method 
 4.4 Solving system of linear equations using Gauss Jordan Elimination  
          Method 
 5. Rank 
  5.1 Rank of a matrix 
  5.2     Linearly Dependent and Linearly Independent Vectors 
  5.3 Row Rank and Column Rank 
  5.4 Consistency of system of linear equations 
 Exercise 
 Summary 
 References 
 
System of Linear Equations and Matrices 
Institute of Lifelong Learning, University of Delhi                                                pg. 3 
 
1. Learning Outcomes 
After Reading this chapter students will be able to understand: 
 ? Homogeneous system of linear equations 
 ? Non-Homogeneous system of linear equations 
 ? The transformations which produces equivalent systems 
 ? Triangular Matrix 
 ? Row reduced echelon Matrix 
? Gaussian Elimination method to transform a system of linear equations 
into an equivalent system in row echelon form 
? Gauss Jordan elimination method to transform a system of linear 
equations into an equivalent system in Reduced row echelon form 
?  Rank, Row Rank and Column Rank of a matrix 
? Solving systems of linear equations using Rank of a matrix 
? Consistency of the system of linear equations. 
 
 
  
Page 4


System of Linear Equations and Matrices 
 
Institute of Lifelong Learning, University of Delhi                                                pg. 1 
 
 
 
 
 
 
 
 
 
 
 
Paper: Linear Algebra 
Lesson: System of Linear Equations and Matrices 
Course Developer: Parvinder Kaur 
Department/College: Assistant Professor, Department of 
Mathematics, Motilal Nehru College, University of Delhi 
 
 
 
 
 
  
System of Linear Equations and Matrices 
Institute of Lifelong Learning, University of Delhi                                                pg. 2 
 
Table of Contents: 
 Chapter : System of Linear Equations and Matrices 
 1. Learning outcomes 
 2. Systems of Linear Equations and Matrices 
  2.1 Introduction 
  2.2 Linear equations 
  2.3 System of Linear equations 
 3. Matrices and Elementary operations 
  3.1 Matrix representation of a linear system of equations 
  3.2 Elementary operations 
  3.3 Elementary matrices 
  3.4 Equivalent systems 
  3.5 Row equivalence  
 4. Special Matrices and their applications 
  4.1 Triangular Matrix 
  4.2 Reduced Row echelon matrix 
 4.3 Solving system of linear equations using Gaussian Elimination   
          Method 
 4.4 Solving system of linear equations using Gauss Jordan Elimination  
          Method 
 5. Rank 
  5.1 Rank of a matrix 
  5.2     Linearly Dependent and Linearly Independent Vectors 
  5.3 Row Rank and Column Rank 
  5.4 Consistency of system of linear equations 
 Exercise 
 Summary 
 References 
 
System of Linear Equations and Matrices 
Institute of Lifelong Learning, University of Delhi                                                pg. 3 
 
1. Learning Outcomes 
After Reading this chapter students will be able to understand: 
 ? Homogeneous system of linear equations 
 ? Non-Homogeneous system of linear equations 
 ? The transformations which produces equivalent systems 
 ? Triangular Matrix 
 ? Row reduced echelon Matrix 
? Gaussian Elimination method to transform a system of linear equations 
into an equivalent system in row echelon form 
? Gauss Jordan elimination method to transform a system of linear 
equations into an equivalent system in Reduced row echelon form 
?  Rank, Row Rank and Column Rank of a matrix 
? Solving systems of linear equations using Rank of a matrix 
? Consistency of the system of linear equations. 
 
 
  
System of Linear Equations and Matrices 
Institute of Lifelong Learning, University of Delhi                                                pg. 4 
 
2. System of Linear Equations and Matrices: 
2.1 Introduction: 
 Linear Algebra is one of the most interesting and applicable areas of 
mathematics. It has wide application in all the areas of mathematics. The theory of 
linear equations plays a significant and motivating role in the subject of linear algebra. In 
fact, many problems in linear algebra are equivalent to studying a system of linear 
equations. The aim of this chapter is to learn about the linear systems and their 
solutions. We will study the notion of a matrix so that we can approach any system via 
its coefficient matrix. This makes us to mention a set of rules called elementary 
operations to transform the system of equations into a reduced row echelon form of the 
system. 
2.2 Linear Equations: 
The linear equation is an expression of the form  
 
1 1 2 2
...
nn
ax a x a x b ? ? ? ?  (1) 
where , ?
i
a b R and '
i
xs are unknowns (or variables). The scalars a
i
 are called the 
coefficient of the '
i
xs respectively and b is called the constant term. 
A set of values for the unknowns, say 
1 1 2 2
, , ... ,
nn
x x x ? ? ? ? ? ? is said to be solution 
of (1) if the statement obtained by substituting 
i
? for 
i
x 
 
1 1 2 2
...
nn
a a a b ? ? ? ? ? ? ? is true. 
This set of values is then said to satisfy the equation we denote this solution by simply 
the n-tuple ? ?
12
, ,..., . ? ? ? ?
n
U 
Value Addition: 
A linear equation can have infinitely many solutions, exactly one solution (Unique 
Solution) or no solution at all. 
 
2.3 System of linear equations 
 A system of linear equations is a finite collection of linear equations in some 
unknowns. For instance, a linear system of m equation in n variables 
12
, ,...,
n
x x x can be 
written as 
Page 5


System of Linear Equations and Matrices 
 
Institute of Lifelong Learning, University of Delhi                                                pg. 1 
 
 
 
 
 
 
 
 
 
 
 
Paper: Linear Algebra 
Lesson: System of Linear Equations and Matrices 
Course Developer: Parvinder Kaur 
Department/College: Assistant Professor, Department of 
Mathematics, Motilal Nehru College, University of Delhi 
 
 
 
 
 
  
System of Linear Equations and Matrices 
Institute of Lifelong Learning, University of Delhi                                                pg. 2 
 
Table of Contents: 
 Chapter : System of Linear Equations and Matrices 
 1. Learning outcomes 
 2. Systems of Linear Equations and Matrices 
  2.1 Introduction 
  2.2 Linear equations 
  2.3 System of Linear equations 
 3. Matrices and Elementary operations 
  3.1 Matrix representation of a linear system of equations 
  3.2 Elementary operations 
  3.3 Elementary matrices 
  3.4 Equivalent systems 
  3.5 Row equivalence  
 4. Special Matrices and their applications 
  4.1 Triangular Matrix 
  4.2 Reduced Row echelon matrix 
 4.3 Solving system of linear equations using Gaussian Elimination   
          Method 
 4.4 Solving system of linear equations using Gauss Jordan Elimination  
          Method 
 5. Rank 
  5.1 Rank of a matrix 
  5.2     Linearly Dependent and Linearly Independent Vectors 
  5.3 Row Rank and Column Rank 
  5.4 Consistency of system of linear equations 
 Exercise 
 Summary 
 References 
 
System of Linear Equations and Matrices 
Institute of Lifelong Learning, University of Delhi                                                pg. 3 
 
1. Learning Outcomes 
After Reading this chapter students will be able to understand: 
 ? Homogeneous system of linear equations 
 ? Non-Homogeneous system of linear equations 
 ? The transformations which produces equivalent systems 
 ? Triangular Matrix 
 ? Row reduced echelon Matrix 
? Gaussian Elimination method to transform a system of linear equations 
into an equivalent system in row echelon form 
? Gauss Jordan elimination method to transform a system of linear 
equations into an equivalent system in Reduced row echelon form 
?  Rank, Row Rank and Column Rank of a matrix 
? Solving systems of linear equations using Rank of a matrix 
? Consistency of the system of linear equations. 
 
 
  
System of Linear Equations and Matrices 
Institute of Lifelong Learning, University of Delhi                                                pg. 4 
 
2. System of Linear Equations and Matrices: 
2.1 Introduction: 
 Linear Algebra is one of the most interesting and applicable areas of 
mathematics. It has wide application in all the areas of mathematics. The theory of 
linear equations plays a significant and motivating role in the subject of linear algebra. In 
fact, many problems in linear algebra are equivalent to studying a system of linear 
equations. The aim of this chapter is to learn about the linear systems and their 
solutions. We will study the notion of a matrix so that we can approach any system via 
its coefficient matrix. This makes us to mention a set of rules called elementary 
operations to transform the system of equations into a reduced row echelon form of the 
system. 
2.2 Linear Equations: 
The linear equation is an expression of the form  
 
1 1 2 2
...
nn
ax a x a x b ? ? ? ?  (1) 
where , ?
i
a b R and '
i
xs are unknowns (or variables). The scalars a
i
 are called the 
coefficient of the '
i
xs respectively and b is called the constant term. 
A set of values for the unknowns, say 
1 1 2 2
, , ... ,
nn
x x x ? ? ? ? ? ? is said to be solution 
of (1) if the statement obtained by substituting 
i
? for 
i
x 
 
1 1 2 2
...
nn
a a a b ? ? ? ? ? ? ? is true. 
This set of values is then said to satisfy the equation we denote this solution by simply 
the n-tuple ? ?
12
, ,..., . ? ? ? ?
n
U 
Value Addition: 
A linear equation can have infinitely many solutions, exactly one solution (Unique 
Solution) or no solution at all. 
 
2.3 System of linear equations 
 A system of linear equations is a finite collection of linear equations in some 
unknowns. For instance, a linear system of m equation in n variables 
12
, ,...,
n
x x x can be 
written as 
System of Linear Equations and Matrices 
Institute of Lifelong Learning, University of Delhi                                                pg. 5 
 
 
11 1 12 2 1 1
...
nn
a x a x a x b ? ? ? ?  (2) 
 
21 1 22 2 2 2
...
nn
a x a x a x b ? ? ? ? 
 ?           ? 
 
1 1 2 2
...
m m mn n m
a x a x a x b ? ? ? ? 
A solution of linear system (2) is a tuple ? ?
12
, ,...,
n
? ? ? of numbers that makes each 
equation a true statement when the values 
12
, ,...,
n
? ? ? are substituted for 
12
, ,...,
n
x x x 
respectively. The set of all solution of a linear system is called the solution set or general 
solution of the system. 
 
System of Linear Equations 
 
 
 
  Consistent     Inconsistent 
 
 
 
Unique solution Infinitely many    No Solution 
        solutions 
 
 
A system of equations is called consistent if there exists a solution otherwise it is called 
as inconsistent. 
The system of linear equations 
 
11 1 12 2 1
... 0
nn
a x a x a x ? ? ? ?  (3) 
 
21 1 22 2 2
... 0
nn
a x a x a x ? ? ? ? 
 ?           ? 
 
1 1 2 2
... 0
m m mn n
a x a x a x ? ? ? ? 
where all b
i
's are zero is called the Homogeneous system of equations. Whereas the 
system of equations (2) is termed as non-Homogeneous system of equation. The