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Explorations of Mathematical Models in Biology with MATLAB by Mazen Shahin

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CHAPTER 3

MODELING WITH MATRICES

3.1. SYSTEMS OF LINEAR EQUATIONS HAVING UNIQUE SOLUTIONS

In this section I introduce the definition of a system of linear equations and its solution(s). I also introduce a method to find the solution of a system of linear equations when the system has a unique solution. The method is called Gauss-Jordan elimination. The Gauss-Jordan elimination method will be introduced in detail in Section 3.2.

Given a system of linear equations, it can be shown that the following operations can be performed to change the system into another system that has the same solution(s) as the original system:

  • Any two equations may be interchanged.
  • Any equation may be multiplied by a nonzero constant.
  • A multiple of an equation may be added to another equation.

These operations are called elementary transformations.

Two systems of linear equations are called equivalent systems, if one system is derived from the other system by applying elementary transformation(s) on it. Equivalently, the two systems have the same solution. I use the symbol ≈ to indicate that two systems are equivalent.

I develop the Gauss-Jordan method to solve a system of linear equations. Example 3.1 illustrates the method that uses elementary transformations.

Example 3.1

Solve the following system of equations

(A)   image

(B)   

(C)   

Solution

We use the first equation to eliminate x from the second equation ...

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