## Column Reduced Echelon Form

A matrix is said to be column reduced if1. The leading entry of each non zero column is 1.

2. Each row containing the leading entry of some non zero column has all its other entries zero.

The following matrices are column reduced.

\[ \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array} \right) , \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right) \]

A matrix is said to be in column reduced echelon form if it satisfies the above two conditions, and also satisfy the following.

3. Each zero column lies to the right of a non zero column.

3. The leading non zero entry in any column is in the row that lies above the leading non zero entry in the next column.

The matrix

\[ \left( \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \end{array} \right) \]

is in column reduced echelon form.A matrix in column reduced echelon form is the transpose of a matrix in row reduced echelon form.