Echelon form of a matrix
A m X n matrix is in row echelon form if it satisfies properties 1, 2, and 3. Furthermore A m X n matrix is in reduced row echelon form if it satisfies the following properties:
1. If there are any zero rows, they must be at the bottom of the matrix.
2. The first nonzero entry from the left of a nonzero row is a 1, which is also called the leading one of that row.
3. The leading one for each nonzero row appears to the right and below any leading ones in the previous rows.
4. For a column with a leading one, the other entries in that column are zero.
A similar definition can be made for reduced column echelon form and column echelon form.
Example
The following matrices are in row echelon form because they follow properties 1, 2, and 3 but not property 4.
$ A=\left[\begin{array}{cccccc}1&4&0&3&-5&-7\\0&1&0&4&2&10\\0&0&0&1&-4&2\\0&0&0&0&0&0\\0&0&0&0&0&0\end{array}\right] $
$ B=\left[\begin{array}{ccccc}1&2&3&4&5\\0&0&1&3&5\\0&0&0&1&-4\\0&0&0&0&0\end{array}\right] $
$ C=\left[\begin{array}{cccc}1&-4&2&8\\0&0&1&-2\\0&0&0&0\end{array}\right] $
The following matrices are in reduced row echelon form because they follow properties 1, 2, 3, and 4.
$ D=\left[\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right] $
$ E=\left[\begin{array}{ccccc}1&7&0&0&3\\0&0&1&-2&6\\0&0&0&0&0\end{array}\right] $
$ F=\left[\begin{array}{cccccc}1&0&0&0&-1&2\\0&1&0&0&3&5\\0&0&0&1&-4&2\\0&0&0&0&0&0\\0&0&0&0&0&0\end{array}\right] $
The following matrices are not in reduced row echelon form.
$ G=\left[\begin{array}{cccc}1&3&0&5\\0&0&0&0\\0&0&1&-7\end{array}\right] $
$ H=\left[\begin{array}{cccc}1&0&7&-4\\0&-4&3&6\\0&0&1&2\end{array}\right] $
$ I=\left[\begin{array}{cccc}1&0&2&3\\0&1&-2&3\\0&1&4&3\\0&0&0&0\end{array}\right] $
$ J=\left[\begin{array}{cccc}1&6&5&2\\0&1&4&3\\0&0&1&-3\\0&0&0&0\end{array}\right] $
Why are these matrices not in reduced row echelon form?
1. Matrix G is not in reduced row echelon form because it violates property 1. Row 2 is a zero row and it is not at the bottom of the matrix.
2. Matrix H is not in reduced row echelon form because it violates properties 2 and 4. Row 2 has the first nonzero entry as a -4 when it needs to be a 1 which violates property 2. Column 3 has a leading one and the other entries in that column are not zero which violates property 4.
3. Matrix I is not in reduced row echelon form because it violates property 3. Row 3 has a leading one directly below the leading one in row 2.
4. Matrix J is not in reduced row echelon form because it violates property 4. Columns 2 and 3 have leading ones and the other entries in that column are not zero.
