Illustration of Gauss elimination:
For illustration, for a 2 × 2 system, an augmented matrix be:
Then, the EROs is applied to obtain the augmented matrix into an upper triangular form:
Therefore, the goal is simply to substitute a_{21} with 0. Here, the primes points that the values (might) have been changed.
Putting this back into the equation form results,
Executing this matrix multiplication for each row outcomes in:
Therefore, the solution is as shown below:
Likewise, for a 3 × 3 system, the augmented matrix is decreased to upper triangular form:
(This will be completed systematically by first obtaining a 0 in the a_{21} position, then a_{31}, and lastly a_{32}.)
Then, the answer will be:
As an illustration, consider the 2 × 2 system of equations as shown below:
As a matrix equation Ax = b, this is:
The initial step is to augment the coefficient matrix A with the b to obtain an augmented matrix [A| b]: