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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.

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
Small Ax=b, det, inverse, rank, LU, Cholesky, QR, SVD, least squares, RREF, condition number, nullspace, column space, rowspace, eigen, multiply, transpose.
Scope
Real 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
  • ISO 80000-2:2019 Quantities and units — Mathematics
  • CRC / Strang — Ax=b, LU/QR/SVD, four fundamental subspaces
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.

2×2 determinantdet = a₁₁a₂₂ − a₁₂a₂₁
SolvePA = LU, then Ly = Pb, Ux = y
iNot a CAS. Solve uses LU with partial pivoting (not explicit A⁻¹). Inverse still uses Gauss–Jordan. Order ≤ 4 square for solve/det/inverse/LU/Cholesky. Cholesky is SPD A=LLᵀ only. QR, SVD, and least squares also accept rectangular A with rows,cols ∈ [2,4]. Inverse of a singular matrix is SINGULAR_MATRIX. Pivot/rank tests are scale-aware (ε·max(m,n)·‖A‖_max). Unique solve publishes residual_2 = ‖Ax−b‖₂ and η = ‖Ax−b‖₂ / (‖A‖₁‖x‖₂ + ‖b‖₂); if the denominator is 0, η = residual_2.

How to use

1

Choose solve, det, inverse, rank, LU, Cholesky, QR, SVD, least squares, RREF, condition number, nullspace, column space, rowspace, left-nullspace, eigen, multiply, or transpose

Solve and least squares need A and rhs b. Multiply needs square A and B of the same order. Det / inverse / rank / LU / Cholesky / QR / SVD / least squares / RREF / condition number / nullspace / column space / rowspace / left-nullspace / eigen also have discovery URLs.

2

Read x, y (and z) or the matrix

Open /workspace/matrix to keep a local matrix scenario.

Example calculations

Common configurations with formula and result.

ϟ

2×2

2x + y = 5, x + y = 3

x=2, y=1
unique
ϟ

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

1·4 − 2·3

−2
−2

Linear System calculator specification

Version 1.14.12 · Engine tested

Calculation status

Review policy · Evidence

Definition
A linear system Ax = b has a unique solution when det(A) ≠ 0. This page is the 2×2 / 3×3 seed of the linear-algebra engine. Larger composition lives at /workspace/matrix.
What it calculates
Small Ax=b, det, inverse, rank, LU, Cholesky, QR, SVD, least squares, RREF, condition number, nullspace, column space, rowspace, eigen, multiply, transpose.
Inputs
  • mode
  • order or rows+cols for lstsq
  • a11…
  • b1…
Outputs
  • x
  • y
  • z
  • det
  • rank
  • eigenvalues
  • singular_values
  • residual_2
  • backward_error
  • L
  • Q
  • status
  • rows
  • cols
Formula
Ax = b
Assumptions
  • Real matrices
  • Order ≤ 4 square for solve/det/inverse/factorizations
  • lstsq also A ∈ R^{m×n} with m,n ∈ [2,4]
  • Planar ℝ² figure when 2×2; other shapes omit it
  • Not a CAS
Units
  • dimensionless
Boundary conditions
  • singular inverse → SINGULAR_MATRIX
  • asymmetric Cholesky → NOT_SYMMETRIC; nonpositive pivot → NOT_POSITIVE_DEFINITE
  • unknown mode (including jordan-form) → INVALID_MODE
Example
2x+y=5, x+y=3 → x=2, y=1
Validation cases

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

  • mode=solve order=2 a11=2 a12=1 a21=1 a22=1 b1=5 b2=3 → x=2 y=1
  • mode=det a11=1 a12=2 a21=3 a22=4 → det=-2
Sources
  • ISO 80000-2:2019 Quantities and units — Mathematics
    Supports: Matrix, determinant, inverse, and rank notation for order 2–4
  • CRC / Strang — Ax=b, LU/QR/SVD, four fundamental subspaces
    Supports: LU + partial pivoting solve; det; Gauss–Jordan inverse; rank-nullity; Householder QR; one-sided Jacobi SVD; RREF; κ₁(A)
Calculation version
1.14.12

Other calculators in this family: Cholesky Factorization Calculator, Column Space Calculator, Condition Number Calculator, Least Squares Calculator, Left Nullspace Calculator, LU Factorization Calculator, Matrix Determinant Calculator, Matrix Eigenvalues 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. › → 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. › → Matrix Eigenvalues Calculator Eigenvalues and eigenvectors of a small real matrix via the characteristic polynomial. Same engine as Linear System. Not a CAS. Runs locally. ›

Frequently asked questions

Key distinctions behind the calculation.

Is this a computer-algebra system?

No. It is small real matrices (order ≤ 4).

Where is the workspace?

/workspace/matrix uses this same engine. Calculator URLs stay /calc/math/linear-system.