What is reduced row echelon form (RREF)?

A matrix is in RREF when: (1) All-zero rows are at the bottom. (2) The first nonzero entry (pivot) in each row is 1. (3) Each pivot is the only nonzero entry in its column. (4) Each pivot is to the right of the pivot in the row above. RREF is unique — every matrix has exactly one RREF. It's found by Gauss-Jordan elimination.

What is the rank of a matrix?

The rank of a matrix is the number of pivot columns in its RREF (equivalently, the number of linearly independent rows or columns). For an m×n matrix, rank ≤ min(m,n). Rank tells you the dimension of the column space and the row space. By the rank-nullity theorem: rank + nullity = n (number of columns).

What is nullity and the rank-nullity theorem?

Nullity = dimension of the null space = number of free variables in the system. The rank-nullity theorem states: rank(A) + nullity(A) = number of columns of A. For example, a 3×5 matrix with rank 2 has nullity 3 — meaning there are 3 free variables and the null space is 3-dimensional.

What are the three elementary row operations?

1) Row swap: interchange two rows. 2) Row scaling: multiply all entries of a row by a nonzero constant. 3) Row addition: replace one row by itself plus a multiple of another row. These operations preserve the row space and hence the rank, making RREF equivalent to the original matrix for solving linear systems.

RREF Calculator — Reduced Row Echelon Form, Rank & Nullity

RREF Calculator

Compute the Reduced Row Echelon Form (RREF) of any matrix with step-by-step row operations. Finds rank, nullity, and pivot columns automatically.

Quick Answer

RREF has leading 1s (pivots) with zeros above and below · Rank = number of pivots · Nullity = columns − rank · Rank-Nullity theorem: rank + nullity = n.

Rows

Columns

Matrix (3×4)

Term	Definition	Meaning
Pivot	Leading 1 in RREF row	Linearly independent direction
Rank	Number of pivot columns	Dimension of column & row space
Nullity	Columns minus rank	Dimension of null space (free variables)
Pivot column	Column containing a pivot	Basis vector for column space
Free column	Non-pivot column	Parameterizes null space vectors
Rank-nullity	rank + nullity = n	Fundamental theorem of linear maps

What Can RREF Tell You?

Solving Linear Systems: Augment the coefficient matrix with the right-hand side [A|b] and row-reduce. Each pivot row gives a solution component; free columns give parameters for infinite solutions.
Linear Independence: Columns are linearly independent iff every column is a pivot column (rank = n for n columns).
Basis for Column Space: The pivot columns of the ORIGINAL matrix (not RREF) form a basis for the column space.
Null Space: Free variable columns give the null space basis vectors directly from the RREF.
Invertibility: An n×n matrix A is invertible iff its RREF is the identity matrix Iₙ (rank = n).

Frequently Asked Questions

What's the difference between REF and RREF?

Row Echelon Form (REF) requires pivots below to be zero, but doesn't require pivots to equal 1 or eliminate entries above. RREF (Reduced REF) goes further: all pivots are exactly 1, and all other entries in pivot columns are 0. RREF is unique; REF is not. Gauss elimination gives REF; Gauss-Jordan gives RREF. Both have the same pivot columns and rank.

Can RREF be used on non-square matrices?

Yes — RREF works on any m×n matrix, not just square ones. A 3×5 matrix in RREF might have 2 pivot rows and 3 free columns. This is one of its most important uses: analyzing underdetermined systems (more unknowns than equations) to find all solutions, or overdetermined systems (more equations than unknowns) to check consistency.

RREF Calculator

What Can RREF Tell You?

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