Row Echelon and Reduced Row Echelon Form¶

Definition¶

A matrix is in row echelon form if all zero rows are at the bottom and each pivot is to the right of the pivot in every row above it.

A matrix is in reduced row echelon form if, in addition, each pivot is $1$ and each pivot column has zeros in all other entries.

Interpretation¶

Row echelon form is a staircase shape that lets a linear system be solved from the bottom up by back substitution. Reduced row echelon form is the fully cleaned-up version obtained by continuing elimination until each pivot column is isolated.

On the left, the blue pivots $2$, $3$ and $5$ form a staircase with grey zeros below them and a zero row at the bottom. On the right is its reduced row echelon form: the pivots equal $1$ and have zeros both above and below. Both matrices have their pivots in columns $1$, $2$ and $4$. Read as the coefficient matrix of $\boldsymbol{A}\boldsymbol{x}=\boldsymbol{0}$$0$, the right one gives $x_1=x_3-2x_5$$2x_5$, $x_2=-2x_3-x_5$ and $x_4=-3x_5$$-3x_5$: each pivot variable is expressed in the free variables $x_3$ and $x_5$, which belong to the columns without a pivot.

The staircase condition is what makes back substitution work: the pivot of each row must lie strictly to the right of the pivot above. The matrix $\begin{pmatrix}0&1\\1&0\end{pmatrix}$ has zeros in two places but is not in row echelon form, since the second pivot lies to the left of the first. Row echelon form is not unique: scaling a row gives another one. The pivot positions, however, do not depend on the row operations used, and the reduced row echelon form of a matrix is unique. The number of pivots, $3$ in the figure, is later called the rank.

Key Results¶

• Every matrix can be brought to row echelon form using elementary row operations.
• The number of nonzero rows in row echelon form equals the rank.
• Reduced row echelon form directly displays the solution structure of a system.

Examples¶

• $\begin{pmatrix}2&1&-2\\0&3&-4\end{pmatrix}$ is in row echelon form.
• $\begin{pmatrix}1&0&-2\\0&1&3\end{pmatrix}$ is in reduced row echelon form.

Prerequisites¶

• matrix
• pivot

Related Concepts¶

• Rank
• Augmented Matrix
• elementary row operations
• Equivalent Systems of Equations
• Solution Set
• System of Linear Equations

Sources¶

• Math II: Linear Algebra · Week 3
• Math V: Advanced Linear Algebra · Week 5 (revisited)

Aliases: REF, RREF, row echelon form, row-reduced echelon form, row canonical form Keywords: echelon form, reduced row echelon form, leading 1

Last updated: 20 October 2026 03:02