A matrix is in **reduced row echelon form** or **RREF** if it is in [[row echelon form]] (REF), so that
- The [[zero rows]] of the matrix (if any) are all at the bottom of the matrix.
- In every non-zero row of the matrix, the [[leading entry]] is $1$.
- If row $i$ and row $(i+1)$ are both non-zero, then the leading entry in row $(i+1)$ is to the right of the leading entry in row $i$. In other words, as you go down the rows, the leading entries must go to the right.
**and** the matrix also has the property:
If a column contains the leading entry of a row, then every other entry in that column is $0$.
For example, \[\begin{bmatrix} {\color{blue}1}&{\color{red}2}&{\color{red}3}&4&5\\0&{\color{blue}1}&{\color{red}2}&3&4\\0&0&{\color{blue}1}&2&3\end{bmatrix}\quad\text{and}\quad \begin{bmatrix} {\color{blue}1}&0&{\color{red}3}&4&5\\0&{\color{blue}1}&0&3&4\\0&0&{\color{blue}1}&2&3\end{bmatrix}\] are both in REF, but they are not in RREF because the red entries are non-zero and are in the same column as a leading entry (in blue).
On the other hand,
\[\begin{bmatrix} {\color{blue}1}&0&0&4&5\\0&{\color{blue}1}&0&3&4\\0&0&{\color{blue}1}&2&3\end{bmatrix}\]
is in RREF.