Example of Gauss-jordan:
For a 2×2 system, this would results
and for a 3 × 3 system,
Note that the resulting diagonal form does not involve the right-most column.
For illustration, the 2 × 2 system, forward elimination results the matrix:
Now, to carry on with back elimination, we require a 0 in the a_{12} position.
Therefore, the solution is x_{1} = 4; -2x_{2} = 2 or x_{2} = -1.
Here is an illustration of a 3× 3 system:
In a matrix form, the augmented matrix [A|b] is as shown below: