Reduced row echelon form

Definition

Let $F$ be a field and let $A$ be a matrix over $F$. The matrix $A$ is in reduced row echelon form when these three conditions hold:

1. $A$ is in row echelon form.
2. The leading entry of each nonzero row is $1$.
3. In each column that has a leading entry, all the other entries are $0$.

Remarks

The matrix $\begin{pmatrix} 1 & 2 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}$ is in reduced row echelon form. Each matrix is row equivalent to exactly one matrix in reduced row echelon form. Gauss–Jordan elimination computes this matrix. The number of nonzero rows of the reduced row echelon form of $A$ is the rank of $A$.

Explanations

L1 human

Reduced row echelon form is the simplest form that row operations can reach. Each pivot is a $1$, and it has zeros above it and below it. From this form we can read the solutions of the system directly.