Gauss Jordan (RREF) elimination for \(Ax=0\) which has a trivial solution. Also known as “a unique solution” or “only one solution”.

So, only one solution (a unique solution) for \(Ax=0\) (Homogeneous system) is called a trivial solution.

❖ To solve a linear system of equations by Gauss Jordan elimination, we have to put the augmented matrix in Reduced Row Echelon Form (RREF).

In this example, the answer to this system has one solution (a unique solution).

(Sometimes, Gauss Jordan elimination is called the Gauss-Jordan reduction method).

❖ The method can process for \(Ax=0\) as the following

\([A | 0 ]\) to \([\text{RREF} | 0 ]\),

So, we have done RREF for the augmented matrix \([ A | 0 ]\).

To illustrate this, you need to convert the linear system into an augmented matrix \([ A | 0 ]\). Then, use matrix row operations to convert the \(4 \times 3\) matrix into the Reduced Row Echelon Form (RREF).

You can easily determine the answers once you convert the augmented matrix to the Reduced Row Echelon Form (RREF).

❖ Previously in this playlist, we have mentioned the steps to determine if a matrix is reduced row echelon form (RREF) or not.

Here, we have explained a unique solution (only one solution) for \(Ax=0\) (Homogeneous system).