RREF Calculator

Compute the Reduced Row Echelon Form of any matrix via Gauss-Jordan elimination. Solve linear systems Ax=b, determine rank and nullity, identify free variables, and apply the Rank-Nullity theorem.

RREF result
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Rank —
Solution —
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RREF
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Rank —
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RREF & Rank

RREF —
Rank —
Nullity —

Rank-Nullity & Free Variables

Rank-Nullity theorem —
Free variables —

How to Use This Calculator

  1. Enter the augmented matrix [A|b] rows — the RREF appears instantly.
  2. Use Solve Ax=b tab for a full 3x3 system.
  3. Use Determine Rank tab to find rank and nullity.
  4. Professional shows free variables and the Rank-Nullity theorem.

Formula

Gauss-Jordan: Apply row operations until each pivot column is a standard basis vector e_i.

Rank-Nullity: rank(A) + nullity(A) = n (number of columns)

Example

[[2,1|5],[4,3|11]] → RREF [[1,0|2],[0,1|1]] → x1=2, x2=1.

Frequently Asked Questions

  • Reduced Row Echelon Form is a matrix where: (1) each pivot is 1, (2) the pivot is the only nonzero in its column, (3) pivots move right and down. It is the unique final form after Gauss-Jordan elimination.
  • Augment [A|b] and reduce to RREF. The right column then gives the solution directly. If a row becomes [0,0,...,0|k] with k nonzero, the system is inconsistent.
  • The rank is the number of pivot rows (nonzero rows) in the RREF. It equals both the row space and column space dimension.
  • For an m×n matrix: rank(A) + nullity(A) = n. Nullity is the number of free variables (non-pivot columns).
  • Row Echelon Form (REF) only requires zeros below pivots. RREF additionally zeroes entries above pivots and scales pivots to 1 — this is the Gauss-Jordan (vs Gaussian) distinction.

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G-J Gauss-Jordan Calculator A-1 Matrix Inverse |A| Determinant Calculator lam Eigenvalue Calculator
Sources & References (5) ▾
  1. Introduction to Linear Algebra — Gilbert Strang — Wellesley-Cambridge Press
  2. OpenStax Linear Algebra — Row Reduction — OpenStax
  3. MIT OCW 18.06 — Elimination with Matrices — MIT OpenCourseWare
  4. Khan Academy — Reduced Row Echelon Form — Khan Academy
  5. Linear Algebra and Its Applications — David C. Lay — Pearson