Gauss-Jordan elimination method in grade IX

Hello friends today we are going to learn the Gauss-Jordan Elimination method. Usually this method we study in grade IX. And also we try to Solve Linear Equations Using Gauss-Jordan Elimination Method. Gauss Jordan is a method to solve the system of linear equation. Gauss Jordan Elimination is the alternative of Gauss Elimination. In Gauss Jordan Elimination we transform the coefficient matrix into the matrix called augmented matrix, which is much easier to solve and gives the same solution as the original set of linear equation and solve the matrix by converting it into unit matrix.
Now question arises what is augmented matrix and unit matrix.

Augmented matrix is the matrix created by adding an additional column for the constants on the right of the equal sign. That is by appending columns of two matrices.
Example:
Suppose we have two matrices A and B, as:
A=1 4 5                       B =  9
     3 7 9           and             8
     6 2 8                              6
Augmented matrix ( A|B ) can be written as:
(A|B) = 1 4 5  | 9
             3 7 9 | 8
             6 2 8 | 6

Unit matrix is the matrix in which all the elements are 0 (zero) expects the diagonal elements. The diagonal elements are 1.
Example:
            1 0 0
            0 1 0
            0 0 1

We perform some elementary operations to solve the equation:

1. Row exchange.
2. Row replacement
3. Row scaling

In order to Solve Linear Equations Using Gauss-Jordan Elimination Method we:

1. represent the system as augmented matrix form.
2. perform elementary row operation on augmented matrix A|B to change it into diagonal form with each diagonal element having a non zero value. If any diagonal element has a zero, switch the row until a non zero value found. If after many efforts we are unable to find some then the system may have either infinite or no solution. Stop then.
3. Divide each diagonal and right hand side element of each row as each diagonal is equal to 1.
4. This form of row is called the reduce echelon form.

Example:
              2b + c=4
            a+b+2c=6
            2a+b+c=7
Step 1:
            Write the system in augmented matrix form.
                        0 2 1 | 4           (R1)
                        1 1 2 | 6           (R2)
                        2 1 1 |7            (R3)
Step 2:
            Perform elementary operations to convert augmented matrix into diagonal form.

                        1 1 2 | 6           (R2)
                        0 2 1 | 4           (R1)
                        2 1 1 | 7           (R3)


                        (R3)+(-2*R1)
                     
                        1   1   2 | 6
                        0   2   1 | 4
                        0 -1  -3 | -5

                        (R1)+(-1/2*R2) and (R3)+(1/2*R2)

                        1  0  3/2    | 4
                        0  2   1      | 4
                        0  0   -5/2  | -3

                        (R1)+(3/5*R3) and (R2)+(2/5*R3)

1    0      0   |  11/5
0    2      0   | -14/5
0    0  -5/2   |  -3

As in two rows we still have values which are not equal to 1. So we now try to convert in into row echelon form.

(R2)*(-1/2) and (R3)*(-2/5)

1   0   0  |  11/5
0   1   0  |   7/5
0   0   1  |   6/5


The right hand side of the augmented matrix is our result. So we have

a=11/5
b=7/5
c=6/5