DIRECT METHOD FOR SOLVING SYSTEMS OF LINEAR EQUATIONS

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  Eliminación de Gauss-Jordan

  
  
  
  
   Suppose it is necessary to find the numbers x, y, z, which multaneously satisfy these equations: 
  
  
  This is called a linear system of equations. The aim is to reduce the equivalent system, which has the same solutions. The operations (called elementary) are these:
  
      * Multiply an equation by a nonzero scalar.
      * Exchange of position two equations
      * Add a multiple of another equation.
  These operations can be represented by elementary matrices used in other procedures such as LU decomposition or diagonalization by congruence of a symmetric matrix.
  In our example, we eliminate x from the second equation by adding 3 / 2 times the first equation to the second and then add the first equation of the third. The result is: 

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Now eliminate the first equation and adding -2 times the second equation to the first, and add -4 times the second equation to eliminate the third y.

 

Finally we eliminate z from the first equation by adding -2 times the third equation to the first, and adding 1 / 2 times the third equation to the second to eliminate z.

 

 Solving, we see solutions:
 

To clarify the steps (and is actually what computers handle), working with the augmented matrix. We see the three steps in its matrix notation:
First:
 

After

Finally.

If the system is incompatible, then we should find a line like this:

Representing the equation 0x + 0y + 0z = 1, ie 0 = 1 has no solution.