Recall that if we have a matrix that is in Row Echelon Form (REF), then we could use Gaussian Elimination, and if necessary, Back Substitution in order to solve a system of linear equations represented by an augmented matrix. We will now look at the similar method of Gauss-Jordan elimination by reducing a matrix to Reduced Row Echelon Form (RREF).

# Gauss-Jordan Elimination

To solve a system by Gauss-Jordan Elimination, we will take a matrix and reduce it fully to RREF. For example consider the following system of linear equations:

(1)

We can represent this system with the following augment matrix:

(2)

When we reduce this matrix to RREF we obtain:

(3)

Note that we can immediately tell the solutions to this system by the matrix, that is $(x_1, x_2, x_3) = (\frac{9}{7}, \frac{4}{7}, 0)$.

## Example 1

**Solve the following system of 3 linear equations of 4 variables given the following matrix $\begin{bmatrix} -9 & -5 & 5 & 3 & 1\\ 7 & 8 & 10 & -3 & -8\\ 5 & 6 & -4 & -9 & 9 \end{bmatrix}$ representing the system using Gauss-Jordan Elimination:**

We will first reduce the augmented matrix of the system to Reduced-Row Echelon Form to obtain the following matrix (we used a computer algebra system (CAS) to obtain the solution since the results are a little messy):

(4)

From this we can see that $x_4$ is our free variable, so let $x_4 = t$ where $t \in \mathbb{R}$. Therefore a general solution arises when:

(5)