Editing Gaussian elimination (section)

=== Echelon form ===
{{Main|Row echelon form}}

For each row in a matrix, if the row does not consist of only zeros, then the leftmost nonzero entry is called the ''[[leading coefficient]]'' (or ''pivot'') of that row. So if two leading coefficients are in the same column, then a row operation of [[#Row operations|type 3]] could be used to make one of those coefficients zero. Then by using the row swapping operation, one can always order the rows so that for every non-zero row, the leading coefficient is to the right of the leading coefficient of the row above. If this is the case, then matrix is said to be in row echelon form. So the lower left part of the matrix contains only zeros, and all of the zero rows are below the non-zero rows. The word "echelon" is used here because one can roughly think of the rows being ranked by their size, with the largest being at the top and the smallest being at the bottom.

For example, the following matrix is in row echelon form, and its leading coefficients are shown in red:
<math display="block">\begin{bmatrix}
  0 & \color{red}{\mathbf{2}} &                     1   & -1 \\
  0 &                     0   & \color{red}{\mathbf{3}} &  1 \\
  0 &                     0   &                     0   &  0
\end{bmatrix}.</math>

It is in echelon form because the zero row is at the bottom, and the leading coefficient of the second row (in the third column), is to the right of the leading coefficient of the first row (in the second column).

A matrix is said to be in reduced row echelon form if furthermore all of the leading coefficients are equal to 1 (which can be achieved by using the elementary row operation of type 2), and in every column containing a leading coefficient, all of the other entries in that column are zero (which can be achieved by using elementary row operations of type 3).