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Math calculator

RREF Calculator

Gauss–Jordan reduced row echelon form for a small real square matrix. Same engine as Linear System. Not a CAS. Runs locally.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes
Input interpretation
Enter values to calculate.
Result
—
Verified scope
all published modes
Assurance
Core
Declared partition coverage
PASS · 19/19 declared partitions (solve, det, inverse, multiply, transpose, rank, eigen, lu, chol, qr, svd, lstsq, rref, condition, nullspace, columnspace, rowspace, leftnullspace, invalid-domain) · Matrix
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
Reduced row echelon form of a small real square matrix (order 2–4).
Scope
Real square matrices
Verification
Engine tested · Source checked · v1.14.12 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes· View Manifest · CVP overview · Specification
Versions
Calculation 1.14.12 · CVP protocol 1.0.0-proposed · Evidence 2026-10-09.qr-root-position-disks
Verification revision
2026-10-09.qr-root-position-disks · 10/10 property · digest 0214764e3fff
Legacy regression
58/58 tests · Production surface contract 5/5
Trust layers
Verification VERIFIED · Production CURRENT · Overall CVP state: VERIFIED under CVP 1.0.0-proposed
Reference
O1 model · O2 expected_values · O2 expected_values · O2 expected_values · O2 expected_values · O2 expected_values · O2 numerical_behavior
Interfaces
PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 10/10 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
431/431 golden · 8/8 CVP boundary · 68/68 invalid · 10/10 property · 1/1 metamorphic · 48/48 cross-interface · 2/2 cross-calculator · 5/5 CVP contract · Manifest
Sources
Sources
Evidence
29 legacy golden · 6 legacy boundary · legacy regression suite · 431/431 oracle-backed golden · 68/68 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.14.12 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

Formulas

Core equations used by this calculator.

RREFA → R (Gauss–Jordan)
Rankrank(A) = number of pivots in R
iSquare matrices only, order ≤ 4. Reuses Gauss–Jordan already used by inverse and rank. Nullspace is a sibling discovery URL. Not a CAS. Float64. Pivot tests are scale-aware (ε·max(m,n)·‖A‖_max).

How to use

1

Enter a square matrix

2×2 or 3×3. Larger composition lives at /workspace/matrix. Rank is a sibling discovery URL.

2

Read R, rank, and pivot columns

pivot_columns are 1-based. Zero rows sit at the bottom.

Example calculations

Common configurations with formula and result.

ϟ

Rank 1

[[1,2],[2,4]]

Gauss–Jordan
[[1,2],[0,0]]
ϟ

Full rank

[[1,2],[3,4]]

R = I
rank=2

RREF calculator specification

Version 1.14.12 · Engine tested

Calculation status

Review policy · Evidence

Definition
Gauss–Jordan elimination produces the unique reduced row echelon form of A, with leading 1s and zeros above and below each pivot. This page is mode=rref on math.linear_algebra. Not a second engine. Nullspace lives on the sibling discovery URL.
What it calculates
Reduced row echelon form of a small real square matrix (order 2–4).
Inputs
  • order
  • a11…
Outputs
  • rref
  • rank
  • pivot_columns
  • det
Formula
Gauss–Jordan → RREF
Assumptions
  • Real square matrices
  • Order ≤ 4
  • Planar ℝ² figure when order=2; order 3+ omits it
  • Float64
  • Not a CAS
  • Not Jordan form / 2-norm κ₂ as a product
Units
  • dimensionless
Boundary conditions
  • unknown mode (including jordan-form) → INVALID_MODE
Example
[[1,2],[2,4]] → [[1,2],[0,0]]
Validation cases

2 published on this page · 58/58 tests · Production surface contract 5/5 · View evidence

  • order=2 a11=1 a12=2 a21=2 a22=4 → rank=1 rref=[[1,2],[0,0]]
  • mode=jordan-form order=2 a11=1 a12=0 a21=0 a22=1 → INVALID_MODE
Sources
Calculation version
1.14.12

Background

Interpretation and common distinctions.

What does this rref calculator compute?

Gauss–Jordan elimination produces the unique reduced row echelon form of A, with leading 1s and zeros above and below each pivot.

The published formula or method is Gauss–Jordan → RREF. The following references support the stated mathematics or method:

Which inputs and limits should you check?

Rank, inverse, subspace, and least-squares decisions depend on the stated tolerance and conditioning. Inspect residuals and warnings before interpreting a nearly singular matrix.

The worked example uses [[1,2],[2,4]] with Gauss–Jordan. Compare it with the Example calculations and the complete result for the same inputs.

How can you verify and cite this result?

CalculatorX is responsible for this page. The mathematical references above explain the model; implementation checks and their scope are documented separately in the math.linear_algebra v1.14.12 Evidence and capability contract. Read the displayed verification and production state together with the specification's numerical limits.

Use the clean calculator URL to cite the tool. A shared input link identifies a particular calculation. Content updates, numerical verification, and domain reviews have separate dates and scopes.

Other calculators in this family: Cholesky Factorization Calculator, Column Space Calculator, Condition Number Calculator, Least Squares Calculator, Left Nullspace Calculator, Linear System Calculator, LU Factorization Calculator, Matrix Determinant Calculator . Explore all Linear Algebra.

→ Cholesky Factorization Calculator Lower Cholesky factorization of a small real symmetric positive-definite matrix. Same engine as Linear System. Not LU. Not a CAS. Runs locally. › → Column Space Calculator Column space (image) of a small real square matrix from RREF pivot columns. Same engine as Linear System. Trivial image is {0}. Left-nullspace is a sibling. Runs locally. › → Condition Number Calculator κ₁(A) = ‖A‖₁ ‖A⁻¹‖₁ for a small real square matrix. Same engine as Linear System. Singular matrices return infinite. Not a 2-norm product. Runs locally. › → Least Squares Calculator Linear least squares for a small real matrix, square or rectangular. Full column rank uses QR; otherwise SVD. Same engine as Linear System. Not Statistics regression. Runs locally. › → Left Nullspace Calculator Left-nullspace (cokernel) of a small real square matrix as ker(Aᵀ). Same engine as Linear System. Trivial left kernel is {0}. Not Jordan form. Runs locally. › → Linear System Calculator Solve small linear systems Ax = b, plus det, inverse, rank, LU, Cholesky, QR, SVD, least squares, RREF, condition number, nullspace, column space, rowspace, left-nullspace, and eigenvalues. Same engine as the matrix workspace. Not a CAS. Runs locally. › → LU Factorization Calculator Doolittle LU factorization with partial pivoting for a small real matrix. Same engine as Linear System. Not a CAS. Runs locally. › → Matrix Determinant Calculator Determinant of a small real matrix. Same engine as Linear System and the matrix workspace. Not a CAS. Runs locally. ›

Frequently asked questions

Key distinctions behind the calculation.

Is this a second linear-algebra engine?

No. It is mode=rref on math.linear_algebra, the same engine as /calc/math/linear-system and /workspace/matrix.

Does this return the nullspace?

No. It returns RREF, rank, and pivot columns. The kernel is /calc/math/nullspace on this same engine.