Jacobi's theorem on determinants
E702054
Jacobi's theorem on determinants is a fundamental result in linear algebra that relates the minors of a matrix to the minors of its adjugate (or inverse), providing key identities used in determinant and matrix theory.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Jacobi's theorem on determinants canonical | 1 |
| Jacobi’s formula for determinants | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7978916 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Jacobi's theorem on determinants Context triple: [Carl Gustav Jacob Jacobi, notableWork, Jacobi's theorem on determinants]
-
A.
Treatise on Demonstration of Problems of Algebra
Treatise on Demonstration of Problems of Algebra is a seminal mathematical work by Omar Khayyam in which he systematically analyzes and geometrically solves cubic equations.
-
B.
Gauss’s remarkable theorem
Gauss’s remarkable theorem is a fundamental result in differential geometry showing that the Gaussian curvature of a surface is an intrinsic property independent of how the surface is embedded in space.
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C.
Theorie der binären algebraischen Formen
"Theorie der binären algebraischen Formen" is a foundational 19th-century mathematical treatise by Alfred Clebsch on the theory of binary algebraic forms and invariants.
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D.
Sylvester determinant
The Sylvester determinant is a mathematical construct introduced by James Joseph Sylvester, typically referring to a determinant associated with resultants and elimination theory in algebra.
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E.
Sylvester’s law of inertia
Sylvester’s law of inertia is a theorem in linear algebra stating that the numbers of positive, negative, and zero eigenvalues (the inertia) of a real symmetric matrix are invariant under change of basis.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Jacobi's theorem on determinants Target entity description: Jacobi's theorem on determinants is a fundamental result in linear algebra that relates the minors of a matrix to the minors of its adjugate (or inverse), providing key identities used in determinant and matrix theory.
-
A.
Treatise on Demonstration of Problems of Algebra
Treatise on Demonstration of Problems of Algebra is a seminal mathematical work by Omar Khayyam in which he systematically analyzes and geometrically solves cubic equations.
-
B.
Gauss’s remarkable theorem
Gauss’s remarkable theorem is a fundamental result in differential geometry showing that the Gaussian curvature of a surface is an intrinsic property independent of how the surface is embedded in space.
-
C.
Theorie der binären algebraischen Formen
"Theorie der binären algebraischen Formen" is a foundational 19th-century mathematical treatise by Alfred Clebsch on the theory of binary algebraic forms and invariants.
-
D.
Sylvester determinant
The Sylvester determinant is a mathematical construct introduced by James Joseph Sylvester, typically referring to a determinant associated with resultants and elimination theory in algebra.
-
E.
Sylvester’s law of inertia
Sylvester’s law of inertia is a theorem in linear algebra stating that the numbers of positive, negative, and zero eigenvalues (the inertia) of a real symmetric matrix are invariant under change of basis.
- F. None of above. chosen
Statements (44)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in linear algebra ⓘ |
| appearsIn |
advanced linear algebra textbooks
ⓘ
treatises on determinant theory ⓘ |
| appliesTo | square matrices ⓘ |
| assumes | matrix is invertible ⓘ |
| field |
determinant theory
ⓘ
linear algebra ⓘ matrix theory ⓘ |
| formalizes | relationship between determinants of complementary submatrices ⓘ |
| gives | identities between minors of a matrix and minors of its adjugate ⓘ |
| hasConcept |
complementary principal minor
ⓘ
index sets of rows and columns ⓘ principal minor ⓘ submatrix ⓘ |
| holdsFor |
complex matrices
ⓘ
matrices over a commutative field ⓘ real matrices ⓘ |
| implies | relations between principal minors and complementary principal minors ⓘ |
| involves |
adjugate matrix
ⓘ
cofactor matrix ⓘ complementary minors ⓘ determinant ⓘ matrix minors ⓘ |
| namedAfter | Carl Gustav Jacob Jacobi ⓘ |
| namedEntityType | theorem ⓘ |
| relatedTo |
Cramer's rule
ⓘ
Laplace expansion of determinants ⓘ adjugate-inverse identity A·adj(A)=det(A)I ⓘ cofactor expansion ⓘ |
| relates |
minors of a matrix
ⓘ
minors of the adjugate matrix ⓘ minors of the inverse matrix ⓘ |
| timePeriod | 19th century mathematics ⓘ |
| usedBy |
engineers working with matrix methods
ⓘ
mathematicians ⓘ theoretical physicists ⓘ |
| usedFor |
computations involving minors
ⓘ
deriving determinant identities ⓘ studying properties of the adjugate matrix ⓘ studying properties of the inverse matrix ⓘ |
| usedIn |
classical invariant theory
ⓘ
multilinear algebra ⓘ theory of linear systems ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Jacobi's theorem on determinants Description of subject: Jacobi's theorem on determinants is a fundamental result in linear algebra that relates the minors of a matrix to the minors of its adjugate (or inverse), providing key identities used in determinant and matrix theory.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject linked to:
Carl
linked to: Jacobi's theorem on determinants