Gauss-Jordan Method

Definitions

The Gauss-Jordan Elimination Method is a technique for solving systems of linear equations of any size.

An equivalent stage is each stage of a sequence of operations on a system of linear equations.

A matrix is a rectangular array of numbers.

A coefficient matrix is the matrix of only the coefficients of an equation.

An augmented matrix is the coefficient matrix augmented by the right-hand side.

An elementary row operation changes the form of a matrix but preserves the solution of the original system.

An equivalent augmented matrix represents an equivalent system.

An example of how to use the Gauss-Jordan method;
The first linear equation.

or;

The second linear equation.

or;

The coefficient matrix consists of the coefficients of the first two columns.

2x	+	4y	=	8
3x	-	2y	=	4

If you drop the x and y variables, you get the augmented matrix of the system.

2	4	8
3	-2	4

Procedure in the equation and matrix forms:
1. Set up the equations like this:

2x	+	4y	=	8		or;		2	4	8
3x	-	2y	=	4		or;		3	-2	4

2. Multiply the first equation by 1/2:

x	+	2y	=	4		or;		1	2	4
3x	-	2y	=	4	or;		3	-2	4

3.A) Multiply the first equation by -3 and add the two equations:

B) You get:

x	+	2y	=	4		or;		1	2	4
		-8y	=	-8		or;		0	-8	-8

C) Multiply the second equation by -1/8:

x	+	2y	=	4		or;		1	2	4
		y	=	1		or;		0	1	1

4.A) Multiply the second equation by -2 and add the two equations:

B) You get:

x			=	2		or;		1	0	2
		y	=	1		or;		0	1	1

5. The following is the solution of the system:

x			=	2		or;		1	0	2
		y	=	1		or;		0	1	1