- The first nonzero number in a row is a 1. (We call it a leading 1).
- All rows of zeros (if there are any) are together at the bottom of the matrix.
- Each column that contains a leading 1, has only zeros below it.
- Each column that contains a leading 1 has zeros everywhere else.

The idea here is that once the matrix is in one of these forms, the system of equations that has this as its augmented matrix can be solved with little or almost no effort.