MST10030 notes wiki

This page is read only. You can view the source, but not change it. Ask your administrator if you think this is wrong.
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 $i$. <html><br /></html>In other words, as you go down the rows, the leading entries must go to the right.

**and** the matrix also has the property:

<html><ol start="4"><li class="level1"><div class="li"></html>
If a column contains the leading entry of a row, then every other entry in that column is $0$.
<html></div></li></ol></html>

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.