Study Notes on Solution of Linear Equations

System of Linear Equations

Introduction

A system of equations is a collection of two or more equations with the same set of unknowns. In solving a system of equations, we try to find values for each of the unknowns that will satisfy every equation in the system. One application of matrices is in solving linear equations (or simultaneous equations as they are often known).

Consider three equations in three unknowns, i.e.

i.e In Matrix From.

The system of three linear equations may be rewritten as

AX=B

If B≠0, the system is called a non-homogeneous system of linear equations.

For |A|≠0, A^-1 exist.

The system becomes AX=B

X=A^-1B

∴ Three linear equations will have a unique solution. In this case, three linear equations are said to be linearly independent.

Important Points:

• Let A be a square matrix. If A is a non-singular matrix, i.e. detA≠0, then the system of linear equations AX=B has a unique solution given by X=A^-1B.
• Let A be a n×n matrix. If detA=0, then the linear system Ax=b has no solution or infinitely many solutions.

For example:

x+y+z=6

2x+3y+4z=20

4x+2y+3z=17

This can be written in terms of matrices:

or more generally

A×X=R

To solve this we simply need to pre-multiply both sides by the inverse of A

A^-1×A×X=A^-1×R

X=A^-1×R

In this case the answer is

Solution of a Homogeneous System of Equations

The solution of non-homogeneous system of n linear equations in n unknowns.

AX=B has 3 possibilities:

(1) If detA≠0, the system has a unique solution.

(2) If detA=0, has no solution or infinitely many solutions.

On the other hand, the homogenous system

AX=0

always has zero solution( trivial solution). Hence, there are only two possible cases for the solution of a system of homogeneous equations:

(1) If detA≠0, the system has only zero solution (trivial solution).

(2) If detA=0, the system has non-zero solutions (non-trivial solution), i.e. it has infinitely many solutions.

For AX=0

x₁=x₂=x₃=0 is the solution of system.

⇒ x₁=x₂=x₃=0 is Trivial Solution

(1) detA≠0 ⇒ A^-1 exist

AX=0

X=A^-10

X=0

∴ the system has trivial solution.

(2) detA=0 ⇒ the system has non-trivial solution.

Solving a 2×2 system of linear equations by using the inverse matrix method

A system of linear equations can be solved by using our knowledge of inverse matrices.

The steps to follow are:

• Express the linear system of equations as a matrix equation.
• Determine the inverse of the coefficient matrix.
• Multiply both sides of the matrix equation by the inverse matrix. In order to multiply the matrices on the right side of the equation, the inverse matrix must appear in front of the answer matrix.(the number of columns in the first matrix must equal the number of rows in the second matrix).
• Complete the multiplication.

The solution will appear as:  where c₁ and c₂ are the solutions.

Examples: Solve the following system of linear equations by using the inverse matrix method:

Solution:  This is the matrix equation that represents the system.

This is the determinant and the inverse of the coefficient matrix.

The common point or solution is (4, -1).

This is the result of multiplying the matrix equation by the inverse of the coefficient matrix.

Solution: 

The common point or solution is (-7, -11).

In the next example, the products will be written over the common denominator instead of being written as two separate fractions.

Solution: 

The common point or solution is (-2, -5).

Solution: 

The common point or solution is (-2, 5)

Exercises: Solve the following systems of linear equations by using the inverse matrix method:

Answers:

Solving systems of linear equations using the inverse matrix method

Cramer's Rule

If |A|≠0, then

Gaussian Elimination

Elementary Transformation

The elementary operations (or elementary transformations) in the process of elimination are:

1. Interchange of two equations;
2. Multiplication of an equation by a non-zero scalar;
3. Addition of a scalar multiple of any equation to another equation.

If the system of equations is written in matrix form, we have the following corresponding elementary row operations for the matrices:

1. Interchange of any rows;
2. Multiplication of any row by a non-zero scalar;
3. Addition of a scalar multiple of any row to another row.

• An inconsistent system of equation is one for which the solution set is the empty set

Example: 

• A consistent system of equations is one for which there exists a non-empty solution set.

Example-1: 

Example-2:  has infinite many solutions

• If the solution set of a consistent system of equations contains one and only one element, the system is said to have unique solution.
• If the solution set of a consistent system of equations contains more than one element, the system is said to have non-unique solution set (infinitely many solution)