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# Gaussian elimination

Tuesday 18 July 2006,

Gaussian elimination is an algorithm in linear algebra for determining the solutions of a system of linear equations. First we do a forward elimination: Gaussian elimination reduces a given system to either triangular. Next, we do a backward elimination to solve the linear system

### Problem

We want to solve the following linear system of equations with unknowns :

In the matrix form, we have

with

### Example of resolution

Consider the following system:

with

First step of the Gaussian elimination: we eliminate in the lines and :

Second step of the Gaussian elimination: we eliminate
in the line :

By backward elimination we solve the linear system, we get the solution :

### Gaussian Elimination algorithm: forward elimination and triangular form

Let the triangular upper matrix, we have

### Gaussian Elimination algorithm: backward elimination

Now the matrix is in triangular form , we can solve:

with the second member after the same operations than .

We use a backward elimination for solving :