Gauss Elimination and Gauss-Jordan Elimination

The elimination method for completing the system of linear equations in the Article Linear Equivalence System in the Form of Matrix in principle is to change the complete matrix of the system of linear equations into another simpler matrix with Elementary Line Operations (OBE). The last form is said to be in the form of a reduced row-echelon form. A matrix n x m is called the reduced row echelon shape if it fulfills the properties:
  1. If a line does not consist entirely of zeros, then the first zero number in that line is 1. (We call this 1 main).
  2. If there is a line consisting entirely of zeros, then all such rows are grouped together under the matrix.
  3. In any two consecutive lines that do not consist entirely of zeros, then 1 main in the lower row is farther to the right than the 1 main in the higher row.
  4. Each column containing a main 1 has zero elsewhere.
A matrix that has properties 1, 2, and 3, is said to be in the form of an echelon form row.

Example 1.

Reduced line echelon matrix,

Looking for a solution to the system of linear equations by performing elementary row operations in a complete matrix to line echelon forms called the Gauss Elmination Method. Whereas if an elementary row operation is carried out only up to the shape of the reduced line echelon is called the Gauss-Jordan Elimination Method.

In the completion of the example below, if the completion process only reaches step 6, problem solving uses the Gauss method. In this case, the completion of a system of linear equations can be obtained by back substitution. If the process below continues until step 8, solving the problem using the Gauss-Jordan method.

Example 2.
In the system of equations:

Step 1,
The enlarged matrix for this system is, 


Step 2, 
Add the first line multiplied by -2 to the second row to get, 


Step 3, 
Add the first line multiplied by -3 to the third row to get,


Step 4,
Multiply the second row with 1/2 to get,


Step 5,
Add the second line multiplied by -3 to the third row to get,


Step 6,
Multiply the third line with -2 to get,

Step 7, 
Add the second line multiplied by -1 in the first row to get,

Step 8,
Add the third line multiplied by -11/2 to the first row and the third row multiplied by 7/2 to the second row to get,


So the solution, we get: x = 1, y = 2, z = 3.


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