Echelon Form

Precise definition

A matrix is in row echelon form (REF) when all zero rows are below nonzero rows, each leading entry of a lower row lies strictly right of the one above, and entries below each leading entry are zero. In reduced row echelon form (RREF), each leading entry is 1 and is the only nonzero entry in its column.

Notation and mathematical language

Leading entries mark pivot positions; corresponding columns in the original coefficient matrix are pivot columns. Nonpivot variable columns correspond to free variables. REF is not unique, but every matrix has a unique RREF.

Conceptual picture

The staircase pattern records which variables are constrained independently. Back substitution solves from REF; RREF displays pivot variables directly in terms of free variables. Row operations preserve the solution set of an augmented system.

Conditions and key results

Pivot columns for a column-space basis must be taken from the original matrix, not from the RREF, because row operations change column vectors. A row $[0\ \cdots\ 0\mid c]$ with $c\ne0$ makes an augmented system inconsistent.

A reliable strategy

1. Use row replacement, row swap, and nonzero row scaling to create the staircase pattern.
2. Identify pivot positions and rank; stop at REF if back substitution suffices.
3. Continue to RREF by scaling pivots and clearing above them when explicit parameter form is helpful.
4. Read consistency, pivot and free variables, then verify solutions in the original equations.

Fully worked example

Interpretation and application

Echelon forms expose consistency, dimension, independence, and rank in one calculation. In numerical computation, pivoting choices affect stability even though exact arithmetic yields the same RREF.

Common mistakes

Verification and reasonableness

• Check each leading position moves right and zeros occur below it.
• Confirm RREF pivots are 1 and alone in their columns.
• Multiply or substitute to verify the solution information against the original matrix.

Practice

1. Is $\begin{pmatrix}1&2\\0&3\end{pmatrix}$ in REF?
2. Is RREF unique?
3. What does a contradictory augmented row mean?

Further deduction

Elementary row operations correspond to left multiplication by invertible elementary matrices. Therefore row-equivalent matrices have the same row space dimension and null space, and augmented systems retain their solution set. Their column spaces are generally different subsets, which is precisely why original pivot columns must be retained for a column-space basis.

Pivot locations also identify four fundamental dimensions. For an $m\times n$ matrix of rank $r$, column and row spaces have dimension $r$, the null space has dimension $n-r$, and the left null space $\ker A^T$ has dimension $m-r$. RREF makes all four counts available from one staircase pattern.

Back-substitution from echelon form gives a unique solution only when every variable column contains a pivot and no contradiction row occurs. For $A:\mathbb R^n\to\mathbb R^m$, a pivot in every row instead means the columns span $\mathbb R^m$, so the map is onto its codomain. That condition does not guarantee uniqueness unless every column also contains a pivot.

Related topics

• Row Reduction
• Gaussian Elimination
• Systems and Matrices