Gauss-Jordan Method

This is a variation of Gaussian elimination. Gaussian elimination gives us tools to solve large linear systems numerically. It is done by manipulating the given matrix using the elementary row operations to put the matrix into row echelon form. To be in row echelon form, a matrix must conform to the following criteria:

1. If a row does not consist entirely of zeros, then the first non zero number in the row is a 1.(the leading 1)
2. If there are any rows entirely made up of zeros, then they are grouped at the bottom of the matrix.
3. In any two successive rows that do not consist entirely of zeros, the leading 1 in the lower row occurs farther to the right that the leading 1 in the higher row.

From this form, the solution is easily(relatively) derived. The variation made in the Gauss-Jordan method is called back substitution. Back substitution consists of taking a row echelon matrix and operating on it in reverse order. Normally the matrix is simplified from top to bottom to achieve row echelon form. When Gauss-Jordan has finished, all that remains in the matrix is a main diagonal of ones and the augmentation, this matrix is now in reduced row echelon form. For a matrix to be in reduced row echelon form, it must be in row echelon form and submit to one added criteria:

• Each column that contains a leading 1 has zeros everywhere else.

Since the matrix is representing the coefficients of the given variables in the system, the augmentation now represents the values of each of those variables. The solution to the system can now be found by inspection and no additional work is required. Consider the following 

example:

Start With:	Elementary Row Operation(S)	Product
	Place into augmented matrix
	R2 - (-1)R1 --> R2 R3 - ( 3)R1 --> R3
	(-1)R2 --> R2 R3 - (-10)R2 --> R3
	(-1/52)R3 --> R3 In Row Echelon Form --->
	R2 - (-5)R3 --> R2 R1 - (2)R3 --> R1
	R1 - (1)R2 --> R1 Reduced Row Echelon Form --->