Applying **Gauss's Method** to any matrix always yields **Echelon Form**.

**Gauss's Method**: If a linear system is changed to another by one of these operations

- an equation is swapped with another
- an equation has both sides multiplied by a nonzero constant
- an equation is replaced by the sum of itself and a multiple of another

then the two systems have the same set of solutions.

**Solution sets** look something like:

$\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$ |

$\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]$ |

$\left[\begin{array}{c}-1/3\\ 2/3\\ 1\end{array}\right]$ z |