Row echelon form

Definition

Let $F$ be a field and let $A$ be an $m \times n$ matrix over $F$. The leading entry of a nonzero row of $A$ is its first nonzero entry, from the left. The matrix $A$ is in row echelon form when these two conditions hold:

1. Each zero row of $A$ is below each nonzero row of $A$.
2. For two consecutive nonzero rows, the leading entry of the lower row is strictly to the right of the leading entry of the upper row.

Remarks

The matrix $\begin{pmatrix} 1 & 2 & 3 \\ 0 & 0 & 4 \\ 0 & 0 & 0 \end{pmatrix}$ is in row echelon form. Some authors also require that each leading entry is $1$. In row echelon form, all the entries below a leading entry are zero. Gaussian elimination changes each matrix into a row echelon form by elementary row operations.

Explanations

L1 human

Row echelon form looks like a staircase. Each nonzero row starts further to the right than the row above it, and the zero rows sit at the bottom. A system in this form is easy to solve from the bottom row up.