Source: https://guthlabs.ai/docs/instruments/matrix-calculator-det-inverse-rref-eigen-least-squares/
instrument / verified-clean
# Matrix Calculator (det/inverse/rref/eigen/least-squares)
The current ledger marks this instrument verified clean within the source-bound verification corpus.
Status: **VERIFIED_CLEAN**Lifecycle: **verified-clean**
## Inspectable records
[01**Overview**Purpose and lifecycle]()[02**Mechanism**Method and assumptions]()[03**Implementation**Binding and validation]()[04**Qualification**Evidence and errata]()
## 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`