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Table of contents
1.
Introduction
2.
System of Linear Equations
3.
Solution of System of Linear Equations
3.1.
Rank of a Matrix 
3.2.
Properties of Rank of a Matrix: 
3.3.
System of Homogeneous Linear Equations
3.4.
System of Non-Homogeneous Linear Equations
4.
FAQs
5.
Key Takeaways
Last Updated: Mar 27, 2024

System of Linear Equations

Author Rajat Agrawal
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Introduction

The theory of linear systems is the foundation and a fundamental portion of linear algebra employed in almost every aspect of modern mathematics. Computational algorithms for solving problems are an important aspect of numerical linear algebra and are used extensively in engineering, physics, chemistry, computer science, and economics. When creating a mathematical model or computer simulation of a highly complex system, a system of nonlinear equations can frequently be approximated by a linear system.

Let’s learn about the system of linear equations in-depth.

System of Linear Equations

A System of Linear Equations is when two or more linear equations are working together. 

Here are some examples of a system of linear equations.

Two linear equations:

<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>x</mi><mo>&#xA0;</mo><mo>+</mo><mo>&#xA0;</mo><mi>y</mi><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>5</mn><mspace linebreak="newline"/><mi>x</mi><mo>&#xA0;</mo><mo>-</mo><mo>&#xA0;</mo><mn>3</mn><mi>y</mi><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>10</mn></math>

Together they are a system of linear equations.

Three linear equations:

<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>+</mo><mi>y</mi><mo>-</mo><mn>3</mn><mi>z</mi><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>5</mn><mspace linebreak="newline"/><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>7</mn><mspace linebreak="newline"/><mn>2</mn><mi>x</mi><mo>&#xA0;</mo><mo>-</mo><mo>&#xA0;</mo><mi>y</mi><mo>&#xA0;</mo><mo>+</mo><mo>&#xA0;</mo><mi>z</mi><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>12</mn></math>

The above three equations are a system of linear equations.

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Solution of System of Linear Equations

There will likely be a solution when the number of equations is the same as the number of variables. Not guaranteed, but likely.

Linear equations may have three kinds of possible solutions: 

1.) No Solution

2.) Unique Solution

3.) Infinitely Many Solution

Consistent System: A System of linear equations having one or more solutions is called a consistent system of equations.

Inconsistent System: A System of equations with no solutions is called an inconsistent system of equations.

Before we move on to finding a solution to a system of linear equations, we will first discuss some introductory algebra.

Rank of a Matrix 

Rank of a matrix is determined as the number of non-zero rows in the row reduced form or the maximum number of independent columns, or the maximum number of independent rows. 

Let A be any mxn matrix, and it has square sub-matrices of different orders. A matrix is said to be of rank r if it satisfies the following properties: 

1.) It has at least one square sub-matrices of order r with a non-zero determinant.

2.) All the determinants of square sub-matrices of order (r+1) or higher than r are zero.

Rank is denoted as P(A). 

Properties of Rank of a Matrix: 

1.) If A is a null matrix, then P(A) = 0, i.e., the Rank of the null matrix is zero.

2.) If In is the nxn unit matrix, then P(A) = n.

3.) The rank of a matrix A mxn, P(A) ≤ min(m,n). Thus P(A) ≤m and P(A) ≤ n.

     P(Anxn ) = n if |A| ≠ 0

4.) If P(A) = m and P(B)=n then P(AB) ≤ min(m,n).

5.) If A and B are square matrices of order n, then P(AB)? P(A) + P(B) – n.

System of Homogeneous Linear Equations

AX = 0

1.) X = 0 is always a solution, and it means all the unknowns have the same value as zero. (This is also known as a trivial solution)

2.) If P(A) = The Number of Unknown Variables, then Unique solution.

3.) If P(A) < The Number of Unknown, Infinite Number of Solutions.

System of Non-Homogeneous Linear Equations

AX = B. 

1.) If P[A: B] ≠P(A), then no Solution.

2.) If P[A: B] = P(A) = The Number of Unknown Variables, then Unique solution.

3.) If P[A: B] = P(A) ≠ The Number of Unknown, then Infinite Number of Solutions.

Here P[A: B] is the rank of gauss elimination representation of AX = B. 

FAQs

1. Define a System of Linear Equations.

A System of Linear Equations is when two or more linear equations are working together.

2. What is a consistent system in the linear equation?

A System of linear equations having one or more solutions is called a consistent system of equations.

3. What is an inconsistent system in the linear equation?

A System of equations with no solutions is called an inconsistent system of equations.

Key Takeaways

In this article, we have extensively discussed the System of Linear Equations, its definition, and its solutions. If you want to learn more, check out our articles on the DeterminantsDoolittle Algorithm, and Partial Differential Equations.

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