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Math calculator

Matrix Determinant Calculator

Determinant of a small real matrix. Same engine as Linear System and 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
det(A) for order 2–3 (engine up to 4).
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
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.

2×2det = a₁₁a₂₂ − a₁₂a₂₁
3×3det = a₁₁(a₂₂a₃₃−a₂₃a₃₂) − a₁₂(a₂₁a₃₃−a₂₃a₃₁) + a₁₃(a₂₁a₃₂−a₂₂a₃₁)
iNot a CAS. Not SVD. Order ≤ 4. Singular matrices have det = 0; inverse is SINGULAR_MATRIX. Eigen / LU / QR share this engine.

How to use

1

Enter a square matrix

2×2 or 3×3. Larger composition lives at /workspace/matrix.

2

Read det(A)

Use Inverse when det ≠ 0, or Rank when you need the dimension of the column space.

Example calculations

Common configurations with formula and result.

ϟ

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

1·4 − 2·3

−2
−2
ϟ

Singular

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

0
0

Matrix Determinant calculator specification

Version 1.14.12 · Engine tested

Calculation status

Review policy · Evidence

Definition
det(A) is the signed volume scale factor of a square matrix. This page is a discovery surface on the linear-algebra engine (mode=det). Inverse, rank, and Ax=b share the same capability.
What it calculates
det(A) for order 2–3 (engine up to 4).
Inputs
  • order
  • a11…
Outputs
  • det
Formula
det(A)
Assumptions
  • Real matrices
  • Order ≤ 4
  • Planar ℝ² figure when order=2; order 3+ omits it
  • Not a CAS
Units
  • dimensionless
Boundary conditions
  • unknown mode (including jordan-form) → INVALID_MODE
Example
[[1,2],[3,4]] → det=−2
Validation cases

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

  • order=2 a11=1 a12=2 a21=3 a22=4 → det=-2
  • order=2 a11=1 a12=2 a21=2 a22=4 → det=0
Sources
Calculation version
1.14.12

Background

Interpretation and common distinctions.

What does this matrix determinant calculator compute?

det(A) is the signed volume scale factor of a square matrix.

The published formula or method is det(A). 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 1·4 − 2·3 with −2. 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 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. › → 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 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 second determinant engine?

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

Where does this run?

Locally in the browser by default. REST and MCP call the same linear-algebra engine.