Gauss-Jordan Elimination

Algorithm: Gauss-Jordan Elimination

The Gauss-Jordan elimination algorithm allows us to determine the solutions of a system of linear equations.

1. Notate the augmented matrix $(A|\vec{b})$.

2. Bring $(A|\vec{b})$ into reduced row echelon form via elementary row operations.

• Make the pivot of the $k$-th row equal to $1$ by multiplying the row with an appropriate constant. Then add an appropriate multiple of the $k$-th row to every row below it in order to obtain only $0$s below its pivot.

3. Examine the solution space of the system:

• If a row of the form $\left[\begin{array}{ccc}0& \cdots & 0\mid \ast \end{array}\right]$, where $\ast \ne 0$, appears at any step of the process, then the system of linear equations has no solutions.

• If the resultant coefficient matrix $A$ has only $1$s on the diagonal and $0$s everywhere else, then system has a unique solution and the last entry in the $k$-th row of the resultant augmented matrix $(A\mid \vec{b})$ is the value for the $k$-th unknown $x_k$.

• If the resultant augmented matrix $(A\mid \vec{b})$ has one or more all-zero rows, then the system has infinitely many solutions.

EXAMPLE

TODO