Reduced Row Echelon Form:

The Gauss Jordan technique results in a diagonal form; for illustration, for a 3 × 3 system:

The Reduced Row Echelon Forms take this one step further to answer in all 1's instead of a', therefore the column of b's is the solution:

The MATLAB has a built-in function to do this, known rref. For illustration, for the preceding illustration:

>> a = [1 3 0; 2 1 3; 4 2 3];

>> b = [1 6 3]';

>> ab = [a b];

>> rref(ab)

ans =

1 0 0 -2

0 1 0  1

0 0 1  3

The answer is found from the last column, therefore x₁ = -2, x₂ = 1, and x₃ = 3. To obtain this in a column vector in a MATLAB:

>> x = ans(:,end)

x =

       -2

        1

        3