{ "category": "Math and trigonometry functions", "display_name": "MINVERSE", "entry_kind": "built_in_function", "eval_api": { "method": "POST", "url": "https://handbook.dnacalc.com/api/eval" }, "function_id": "FUNC.MINVERSE", "implementation_state": { "answer_bank": { "excel_builds": [ "20131", "20228" ], "larger_numeric": 0, "last_bit": 0, "matched": 1683, "other_mismatch": 0, "refused": 0, "rows": 1795, "unread": 112, "worst_ulp": 0 }, "detail": "Recorded: 1,683 in the answer bank and 68 workbook cells.", "dnacalc_commit": "9e1884369a678158dd82d3d74188daf986995e87", "in_catalogue": true, "state": "matches", "state_label": "Matches every recorded Excel answer", "summary": "Matches all 1,751 recorded answers", "workbook_corpus": { "cells": 68, "matched": 68 } }, "implementations": [ { "commit": "9e1884369a678158dd82d3d74188daf986995e87", "flavour": "excel-compatibility-candidate", "language": "Rust", "name": "CalcFunc", "own_code": [ "https://github.com/DnaCalc/DnaCalc/blob/9e1884369a678158dd82d3d74188daf986995e87/crates/calc-func/src/catalogue/math.rs#L222-L226", "https://github.com/DnaCalc/DnaCalc/blob/9e1884369a678158dd82d3d74188daf986995e87/crates/calc-func/src/ported/matrix_family.rs#L59-L130" ], "repository": "https://github.com/DnaCalc/DnaCalc", "shared_code": [ "catalogue/math::matrix", "ported/excel_numeric::binary_with", "ported/excel_numeric::excel_x87_sub", "ported/excel_numeric::excel_x87_mul", "ported/excel_numeric::excel_x87_div", "ported/excel_numeric/x87::mul", "ported/excel_numeric/x87::sub", "ported/excel_numeric/x87::div" ], "source": "https://github.com/DnaCalc/DnaCalc/blob/9e1884369a678158dd82d3d74188daf986995e87/crates/calc-func/src/catalogue/math.rs#L222-L226", "state": "matches", "suite": null, "summary": "Matches all 1,751 recorded answers" } ], "json_url": "https://handbook.dnacalc.com/api/functions/FUNC.MINVERSE.json", "name": "MINVERSE", "notes": [], "schema": "efh.function-reference/v1", "section_order": [ "what-it-does", "try-it", "quirks", "how-to-implement", "version-differences", "alternatives" ], "sections": { "alternatives": "\n### Relationships\n\n\n- **[MMULT](FUNC.MMULT.md)** — the verifier. `MMULT(A, MINVERSE(A))` against\n [MUNIT](FUNC.MUNIT.md) is the residual test, and it is the single most useful thing a reader can\n do with these four functions together.\n- **[MDETERM](FUNC.MDETERM.md)** — shares the family's LU forward elimination in the reference\n engine, and is related mathematically by det(A⁻¹) = 1/det A. **Sharing an elimination is not\n sharing a measurement**: `MDETERM`'s own evidence record forbids it from borrowing this page's\n figures, and this page does not lend them.\n- **[MUNIT](FUNC.MUNIT.md)** — the identity the inverse is defined against.\n- **`LINEST` / `TREND` / `LOGEST`** — regression surfaces that solve linear systems internally.\n They are the reason most worksheets *think* they need `MINVERSE`, and they are usually the better\n tool, because they solve rather than invert.\n- **`SUMPRODUCT` with an explicit 2×2 formula** — for the smallest cases, writing out\n `1/(ad-bc)·[d,-b;-c,a]` is a legitimate alternative and is often more accurate than a general\n factorisation, because it performs fewer operations.\n\n**The most important relationship is a negative one: do not invert to solve.** To compute A⁻¹b,\nsolve Ax = b. Forming the inverse and multiplying is slower, less accurate, and needs a\nnonsingular matrix where a solve needs only a consistent system. Higham devotes a section to this\n(*Accuracy and Stability of Numerical Algorithms*, chapter 14, on the \"fallacy\" of matrix\ninversion); Excel's worksheet surface unfortunately makes the wrong route the easy one, since\n`MINVERSE` exists and a general solve does not.\n\n\n", "how-to-implement": "\n### Numerical notes\n\n\n### The algorithm space, and what has been ruled out of it\n\n- **Adjugate / cofactor (Cramer's rule)** — ruled out at both sizes tested. Cramer's rule is the\n textbook formula and is numerically the worst of the candidates; it is also O(n!) if implemented\n literally.\n- **An x87 80-bit extended-precision body with double stores** — ruled out, and strictly worse than\n plain binary64. Note the contrast with [LN](FUNC.LN.md) and [LOG](FUNC.LOG.md), where the x87\n substrate is exactly what *was* identified. Different Excel surfaces sit on different substrates,\n and that is a finding in itself.\n- **A systematic sweep of solve orderings** — forward against back substitution, streamed against\n sum-then-subtract accumulation, ascending against descending index order, division against\n reciprocal-multiply — all eliminated. This is the shape of the last-bit problem: the *op graph*,\n not the algorithm name, is what determines the final bits.\n\n\n### Why the last bits are hard at all\n\n### Conditioning governs everything\n\nThe relative error in a computed inverse is bounded by roughly κ(A)·u, with κ the condition number\nand u the unit roundoff. For an ill-conditioned matrix the computed inverse can have **no correct\ndigits at all** while still satisfying a small backward error. Two consequences a reader should\ncarry away:\n\n1. **A small residual `MMULT(A, MINVERSE(A)) − I` does not certify the inverse.** Backward\n stability gives you a small residual almost regardless of conditioning; it does not give you a\n small forward error.\n2. **Near-singular is a continuum, not a boundary.** The `#NUM!` from `MINVERSE` fires at some\n threshold; just inside it, the function returns numbers that are formally defined and\n practically meaningless. The dangerous inputs are the ones that do *not* error.\n\nGolub and Van Loan, *Matrix Computations*, section 3.5, and Higham chapter 14, are the standard\ntreatments; the residual-versus-error distinction is Higham's chapter 7.\n\n\n", "quirks": "\n### Result and edge cases\n\n\nReturns `Array` of `Number`.\n\n- **A 1×1 zero matrix** produces `#NUM!` — singular, correctly.\n- **A 1×1 matrix holding −1** returns a 1×1 array holding −1, its own reciprocal.\n- **A logical argument** and **numeric text** both produce `#VALUE!`. As with\n [MDETERM](FUNC.MDETERM.md), the matrix family refuses conversions that the scalar math functions\n accept. The text refusal is documented; the logical refusal is not.\n- **The largest finite double** as a 1×1 matrix returns its reciprocal, a subnormal-range value.\n- **The smallest positive subnormal** as a 1×1 matrix produces **`#NUM!`** — not a reciprocal.\n Mathematically the matrix is nonsingular; its inverse simply is not representable, since the\n reciprocal overflows. The engine reports it as the singular case. **The documented criterion is\n \"the determinant is 0\", and this matrix's determinant is not zero**, so the observable behaviour\n and the documented rule part company here. Whatever the right design is, the honest statement is\n that `#NUM!` from `MINVERSE` covers at least two distinct conditions — genuinely singular, and\n representably impossible — and only one of them is documented.\n- **An inline 2×2 integer array** returns an inverse whose entries **are not the exactly\n representable values the true inverse has**. The exact inverse of that particular matrix has\n entries every one of which is a binary64 value; the returned entries differ from them in the last\n bits. This is not a defect report against anyone — it is the expected signature of a\n factorisation-based solve, and it is exactly why this function has a live numeric investigation\n attached. See \"Numerical notes\".\n\n\n\n### Errors\n\n\n| Error | Documented condition |\n|---|---|\n| `#VALUE!` | \"If any cells in array are empty or contain text, MINVERSE returns a #VALUE! error\" |\n| `#VALUE!` | \"MINVERSE also returns a #VALUE! error if array does not have an equal number of rows and columns\" |\n| `#NUM!` | The matrix cannot be inverted; \"The determinant for a noninvertable matrix is 0\" |\n\nError values inside the array propagate; the family declares a reduction error-collapse profile,\nso competing errors are folded rather than surfacing independently\n([coercion and lifting](../model/02-coercion-and-lifting.md)).\n\n**On the documented `#NUM!` criterion.** \"The determinant is 0\" is the mathematical criterion and\nis not a computational test — see the singularity discussion on\n[MDETERM](FUNC.MDETERM.md). No implementation can determine exact singularity from rounded inputs;\nwhat it can do is detect a pivot that has become negligible relative to the matrix's scale. So the\ndocumented rule describes the *intent*, and the observed threshold behaviour is a different, and\nundocumented, thing. The subnormal case above shows the gap is real.\n\n**On the accuracy remark.** Microsoft's page states:\n\n> \"MINVERSE is calculated with an accuracy of approximately 16 digits, which may lead to a small\n> numeric error when the cancellation is not complete.\"\n\nAs on [MDETERM](FUNC.MDETERM.md), this is not an error bound: \"approximately 16 digits\" describes\nbinary64 generally, and \"when the cancellation is not complete\" names no determinable condition.\nThe real bound is conditioning-dependent and is stated below.\n\n\n", "sources": "- Microsoft, \"MINVERSE function\" —\n .\n Retrieved for this page: the syntax, the array argument description and its permitted forms, the\n two `#VALUE!` conditions, the `#NUM!` singular condition with its determinant criterion, the\n accuracy remark quoted above, and the dynamic-array versus legacy-array entry mechanics.\n- Handbook evidence record `EV-STRUCT-0012` — provisional structural COM probe over the matrix\n family; pinned values, no count, no named build.\n- N. J. Higham, *Accuracy and Stability of Numerical Algorithms*, 2nd edition, chapters 7 and 14 —\n residual versus forward error, LU error analysis, and the case against forming inverses.\n- G. H. Golub and C. F. Van Loan, *Matrix Computations*, 4th edition, sections 3.2 and 3.5 —\n pivoting, conditioning, and solving in preference to inverting.\n- Handbook, [MDETERM](FUNC.MDETERM.md), [MMULT](FUNC.MMULT.md), [MUNIT](FUNC.MUNIT.md) — the rest\n of the family; [coercion and lifting](../model/02-coercion-and-lifting.md) — error propagation\n and the reduction fold.\n\n\n", "version-differences": "The recorded environments below distinguish a reading from a same-input comparison. Counts describe captured inputs in each environment; overlapping captures and comparisons are not totals of unique worksheet calls.\n\n| Excel 16.0 build | Platform and CPU | Captured inputs |\n|---|---|---:|\n| 20131 | Windows, bitness not recorded, AMD Ryzen 7 4800H with Radeon Graphics | 49 |\n| 20228 | Windows, 64-bit, CPU not recorded | 985 |\n\nThe record contains no same-input comparison between these environments. Reading different inputs on different builds does not establish either a version change or cross-build agreement.\n\n[Recorded environments and comparison details](https://github.com/DnaCalc/ExcelFunctionsHandbook/blob/5dc75391aa5125ccc5abdb095da85950980d6fd8/content/evidence/environments/FUNC.MINVERSE.json).", "what-it-does": "\n`MINVERSE(array)` returns the **matrix inverse** of a square matrix: the unique matrix A⁻¹ with\n\n A · A⁻¹ = A⁻¹ · A = I\n\n**Existence.** A⁻¹ exists exactly when *A* is nonsingular, equivalently when det A ≠ 0,\nequivalently when the columns are linearly independent, equivalently when the only solution of\nAx = 0 is x = 0. Microsoft's page states the criterion in the determinant form: \"Some square\nmatrices cannot be inverted and will return the #NUM! error value with MINVERSE. The determinant\nfor a noninvertable matrix is 0.\"\n\n**The closed form** — and the reason it must not be used as an algorithm — is Cramer's rule:\n\n A⁻¹ = adj(A) / det(A)\n\nwhere adj(A) is the transpose of the cofactor matrix. For a 2×2 matrix this is the familiar\n\n [a b; c d]⁻¹ = 1/(ad − bc) · [d −b; −c a]\n\n**Properties** worth carrying: (AB)⁻¹ = B⁻¹A⁻¹ (note the order reversal); (Aᵀ)⁻¹ = (A⁻¹)ᵀ;\ndet(A⁻¹) = 1/det(A); and the inverse of a diagonal matrix is the elementwise reciprocal. The map\nA ↦ A⁻¹ is smooth on the open set of nonsingular matrices, with derivative −A⁻¹ (dA) A⁻¹ — which\nis the analytic statement of the fact that inversion is badly behaved near singularity: the\nderivative blows up as A⁻¹ does.\n\n**Domain and range.** Domain: nonsingular square numeric matrices. Range: nonsingular square\nmatrices. There is no pole in the analytic sense, but the set of singular matrices is a\nmeasure-zero hypersurface (the zero set of the determinant polynomial) across which the function\nis unbounded — the closest thing to a singularity a matrix function has, and the origin of every\ndifficulty below.\n\n\n\n### Arguments\n\n\n| Argument | Meaning | Required |\n|---|---|---|\n| `array` | \"A numeric array with an equal number of rows and columns.\" | Yes |\n\nExactly one argument; the declared arity is one to one. Microsoft's page notes the array may be\ngiven as a cell range such as `A1:C3`, an array constant such as `{1,2,3;4,5,6;7,8,9}`, or a name.\n\n**Result shape.** `MINVERSE` returns an array the same size as its input. Microsoft's page states\nthe entry mechanics explicitly: in current Microsoft 365 the formula is entered in the top-left\ncell and confirmed with Enter, spilling; in earlier versions the output range must be selected\nfirst and the formula confirmed with Ctrl+Shift+Enter as a legacy array formula. That difference\nis a property of the host, not of the function.\n\n\n" }, "signature": { "display": "MINVERSE(array)", "parameters": [ { "description": null, "name": "array", "optional": false, "repeats": false } ], "trailing_repeats": false }, "sources": { "metadata": { "admission": { "label": "implemented", "reason": null }, "arity": { "max": 1, "min": 1 }, "category": "Math and trigonometry functions", "classification": { "arg_preparation_profile": "RefsVisibleInAdapter", "coercion_lift_profile": "Custom", "determinism_class": "Deterministic", "error_collapse_profile": "ReductionFold", "fec_dependency_profile": "RefOnly", "host_interaction_class": "None", "kernel_signature_class": "Custom", "rich_value_usage": "RichBlind", "surface_fec_dependency_profile": "RefOnly", "thread_safety_class": "SafePure", "volatility_class": "NonVolatile" }, "descriptions": { "long": null, "short": "Inverse of a square matrix", "short_source": "dna-calc-own" }, "docs": { "english_description": "Returns the matrix inverse of an array", "microsoft_url": "https://support.microsoft.com/en-us/office/minverse-function-11f55086-adde-4c9f-8eb9-59da2d72efc6" }, "entry_kind": "built_in_function", "localized_family_names": null, "localized_names": [ { "locale": "ar-SA", "name": "MINVERSE" }, { "locale": "bg-BG", "name": "MINVERSE" }, { "locale": "cs-CZ", "name": "MINVERSE" }, { "locale": "da-DK", "name": "MINVERT" }, { "locale": "de-DE", "name": "MINVERSE" }, { "locale": "el-GR", "name": "MINVERSE" }, { "locale": "en-AU", "name": "MINVERSE" }, { "locale": "en-GB", "name": "MINVERSE" }, { "locale": "en-US", "name": "MINVERSE" }, { "locale": "es-ES", "name": "MINVERSA" }, { "locale": "et-EE", "name": "MINVERSE" }, { "locale": "fi-FI", "name": "MKÄÄNTEINEN" }, { "locale": "fr-FR", "name": "INVERSEMAT*" }, { "locale": "he-IL", "name": "MINVERSE" }, { "locale": "hr-HR", "name": "MINVERSE" }, { "locale": "hu-HU", "name": "INVERZ.MÁTRIX" }, { "locale": "id-ID", "name": "MINVERSE" }, { "locale": "it-IT", "name": "MINVERSE" }, { "locale": "ja-JP", "name": "MINVERSE" }, { "locale": "ko-KR", "name": "MINVERSE" }, { "locale": "lt-LT", "name": "MINVERSE" }, { "locale": "lv-LV", "name": "MINVERSE" }, { "locale": "nb-NO", "name": "MINVERSE" }, { "locale": "nl-NL", "name": "INVERSEMAT" }, { "locale": "pl-PL", "name": "MACIERZ.ODW" }, { "locale": "pt-BR", "name": "MINVERSE" }, { "locale": "pt-PT", "name": "MATRIZ.INVERSA" }, { "locale": "ro-RO", "name": "MINVERSE" }, { "locale": "ru-RU", "name": "МОБР" }, { "locale": "sk-SK", "name": "MINVERSE" }, { "locale": "sl-SI", "name": "MINVERSE" }, { "locale": "sr-Latn-RS", "name": "MINVERSE" }, { "locale": "sv-SE", "name": "MINVERT" }, { "locale": "th-TH", "name": "MINVERSE" }, { "locale": "tr-TR", "name": "DİZEY_TERS işlevi" }, { "locale": "uk-UA", "name": "MINVERSE" }, { "locale": "vi-VN", "name": "MINVERSE" }, { "locale": "zh-CN", "name": "MINVERSE" }, { "locale": "zh-HK", "name": "MINVERSE" }, { "locale": "zh-TW", "name": "MINVERSE" } ], "signature": { "display": "MINVERSE(array)", "parameters": [ { "description": null, "name": "array", "optional": false, "repeats": false } ], "trailing_repeats": false }, "surface_name": "MINVERSE", "version_marker": null, "xlcall": { "code": "164", "symbol": "xlfMinverse" } }, "microsoft": "https://support.microsoft.com/en-us/office/minverse-function-11f55086-adde-4c9f-8eb9-59da2d72efc6" }, "try_it": { "formula": "=MINVERSE({1,2;3,4})" }, "url": "https://handbook.dnacalc.com/functions/MINVERSE.html", "variants": [], "versions": [ { "behaviour": "49 captured inputs are indexed for this environment. Capture dates: 2026-07-12 to 2026-07-12. This is a reading count, not a comparison with another build.", "excel_build": "16.0 build 20131", "platform": "Windows, bitness not recorded, AMD Ryzen 7 4800H with Radeon Graphics", "sources": [ "https://github.com/DnaCalc/ExcelFunctionsHandbook/blob/5dc75391aa5125ccc5abdb095da85950980d6fd8/content/evidence/environments/FUNC.MINVERSE.json" ] }, { "behaviour": "985 captured inputs are indexed for this environment. Capture dates: 2026-08-09 to 2026-08-09. This is a reading count, not a comparison with another build.", "excel_build": "16.0 build 20228", "platform": "Windows, 64-bit, CPU not recorded", "sources": [ "https://github.com/DnaCalc/ExcelFunctionsHandbook/blob/5dc75391aa5125ccc5abdb095da85950980d6fd8/content/evidence/environments/FUNC.MINVERSE.json" ] } ] }