Matrix Calculator (det/inverse/rref/eigen/least-squares)

Inspectable records

Ledger state

The current ledger marks this instrument verified clean within the source-bound verification corpus.

Area

Mathematics — arithmetic through advanced (algebra, matrices, calculus, sequences/series, number theory, combinatorics, graphing, complex numbers, geometry, logic, sets)

Tranche

tranche-8

Evidence ID

t8-019

Ledger rows

1

No ledger note is attached.

Review findings

The elementary formulas are correct on well-conditioned, dimensionally valid inputs, but the stated numerical algorithms and unspecified failure policies do not support the full universal claim. The honest label appropriately warns about eigen-solver instability, but it should also disclose least-squares instability from normal equations, possible complex eigenvalues, tolerance dependence, and undefined behavior for singular or rank-deficient cases.

Formula and scope findings

• LU decomposition is a valid determinant method only when pivoting is handled correctly; row swaps must change the determinant sign, and singular or nearly singular pivots require explicit handling.
• Cofactor expansion is mathematically correct but is computationally impractical for large variable-size matrices and should not be presented as a general-purpose implementation method.
• A plain QR eigenvalue algorithm is not sufficient as a universal eigen solver: it needs convergence controls, usually shifts or a Hessenberg/Schur reduction, and support for complex conjugate eigenvalues. It may fail or return inaccurate results for repeated, defective, or ill-conditioned matrices.
• Normal equations A^T A beta=A^T b are mathematically valid for full-column-rank least squares, but they square the condition number and can produce materially inaccurate results for ill-conditioned inputs. QR or SVD should be the robust default.
• The specification does not define behavior for singular inverse requests, rank-deficient or underdetermined least-squares systems, complex eigenvalues, eigenvalue multiplicities, convergence failure, or numerical tolerance selection.
• Matrix addition and multiplication require compatible dimensions, but those constraints and the behavior on dimension mismatch are not specified.

Unit assumptions

• Matrix entries are treated as dimensionless numbers unless the caller supplies a consistent physical interpretation.
• Matrix addition requires corresponding entries to have compatible units; matrix multiplication requires inner dimensions to match and produces compound units.
• A determinant of an n by n matrix has units equal to the product of the units of the selected entries; an inverse has reciprocal units in the corresponding linear-map sense.
• Least-squares rows must use compatible units for the response vector, and columns of the design matrix must be scaled consistently before interpreting coefficients.

Boundary hardening

Recorded hazards

• Inverse has a division-by-zero hazard when det(A)=0; near-zero determinants require a tolerance policy rather than an exact floating-point comparison.
• LU, QR, RREF, eigenvalue, rank, and least-squares results depend on pivoting, scaling, floating-point tolerances, and iteration limits.
• Normal equations can lose significant digits for ill-conditioned design matrices; rank-deficient systems can make A^T A singular.
• Eigenvectors are non-unique up to scaling, and repeated or defective eigenvalues may not have a complete eigenbasis.
• Real-input matrices can have complex eigenvalues and eigenvectors; the output type and supported arithmetic are unspecified.
• Variable-size inputs need explicit maximum dimensions and clear errors for malformed, ragged, empty, non-square, or dimensionally incompatible matrices.
• An offline build that needs a vendored math.js library must verify that the vendored version supports all claimed operations and has no runtime network dependency.

Errata and corrections

No applied erratum is attached to this unique-name record.

Inspectable example

4 review vectors are recorded. One source-bound sample is shown below.

{
  "inputs": {
    "operation": "determinant_and_inverse",
    "A": [
      [
        4,
        7
      ],
      [
        2,
        6
      ]
    ]
  },
  "expected_outputs": {
    "determinant": "10",
    "inverse": [
      [
        "3/5",
        "-7/10"
      ],
      [
        "-1/5",
        "2/5"
      ]
    ],
    "verification_product": [
      [
        1,
        0
      ],
      [
        0,
        1
      ]
    ],
    "spot_checks": [
      {
        "at": "det([[4,7],[2,6]])",
        "input_side": 10,
        "answer_side": 10
      }
    ]
  },
  "reference_data": {
    "method": "For a 2 by 2 matrix [[a,b],[c,d]], det=ad-bc and inverse=(1/(ad-bc))*[[d,-b],[-c,a]]."
  },
  "working": "det(A)=4*6-7*2=24-14=10. Therefore inverse=(1/10)*[[6,-7],[-2,4]]=[[6/10,-7/10],[-2/10,4/10]]=[[3/5,-7/10],[-1/5,2/5]]. Product check: first row, first column=4*(3/5)+7*(-1/5)=12/5-7/5=1; first row, second column=4*(-7/10)+7*(2/5)=-28/10+28/10=0; second row, first column=2*(3/5)+6*(-1/5)=6/5-6/5=0; second row, second column=2*(-7/10)+6*(2/5)=-14/10+24/10=1."
}

Related workflows

No source-defined multi-instrument workflow membership is recorded.

Sources and evidence

These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.

• VERIFIED_LEDGER.json · VERIFIED_CLEAN
• evidence-tranche8/tranche8-results.json