Gauss-Jordan Elimination

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)
\begin{align} 3x_1 + 2x_2 + x_3 = 5 \\ 2x_1 - x_2 + 3x_3 = 2 \\ x_1 + 3x_2 - x_3 = 3 \end{align}

We can represent this system with the following augment matrix:

(2)
\begin{bmatrix} 3 & 2 & 1 & 5\\ 2 & -1 & 3 & 2\\ 1 & 3 & -1 & 3 \end{bmatrix}

When we reduce this matrix to RREF we obtain:

(3)
\begin{bmatrix} 1 & 0 & 0 & \frac{9}{7}\\ 0 & 1 & 0 & \frac{4}{7} \\ 0 & 0 & 1 & 0 \end{bmatrix}

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)
\begin{bmatrix} 1 & 0 & 0 & \frac{9}{8} & -\frac{491}{224}\\ 0 & 1 & 0 & -\frac{33}{16} & \frac{1115}{448}\\ 0 & 0 & 1 & \frac{9}{16} & -\frac{563}{448} \end{bmatrix}

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)
\begin{align} \quad x_1 = -\frac{491}{224} - \frac{9}{8}t \: , \quad x_2 = \frac{1115}{448} + \frac{33}{16}t \: , \quad x_3 = -\frac{563}{448} - \frac{9}{16}t \: , \quad x_4 = t \end{align}