Definition of matrix in reduced row echelon form

Definition: A matrix is in reduced row echelon form (RREF) if:

1. It is in row echelon form.
2. Every leading entry is 1 (these are called leading 1s).
3. Each leading 1 is the only nonzero entry in its column.

Example:
$$\begin{pmatrix} 1 & 0 & 0 & 3 \\ 0 & 1 & 0 & -2 \\ 0 & 0 & 1 & 5 \end{pmatrix}$$

This is in RREF: each leading 1 is the only nonzero entry in its column.

Contrast with REF: A matrix in (plain) row echelon form may have nonzero entries above pivots and pivots other than 1. RREF adds the uniqueness property.