Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
If is a unit, then
Statement
If is a commutative ring, , , and is a unit, then
Facts & Assumptions
Given: as in the statement.
The adjugate identity is (For every positive-sized square matrix over a commutative ring, ).
A positive-sized square matrix whose determinant is a unit is invertible (A positive-sized square matrix over a commutative ring is invertible if and only if its determinant is a unit).
A two-sided inverse in a monoid is unique (In a monoid, a left inverse and a right inverse of the same element are equal; hence an invertible element has exactly one inverse, and it is two-sided).
Proof
Put . Scaling [L1] by gives .
By [L2], exists. Both it and are two-sided inverses of , so [L3] makes them equal.
Depends on
- For every positive-sized square matrix over a commutative ring, $A\operatorname{adj}(A)=\operatorname{adj}(A)A=\det(A)I$
- A positive-sized square matrix over a commutative ring is invertible if and only if its determinant is a unit
- In a monoid, a left inverse and a right inverse of the same element are equal; hence an invertible element has exactly one inverse, and it is two-sided
Used by
- Multiplication by 2 on ℤ is injective but not surjective: its determinant is the non-unit 2, its adjugate is integral, and its inverse exists after extending scalars to ℚ Example
- The adjugate gives the inverse of a 3×3 rational matrix with determinant 3 Example
- The special linear group is a codimension-one embedded submanifold Example
- Unitary and special unitary Lie groups Example
- Matrix inversion preserves Cᵏ regularity where the determinant is nonzero Lemma
- Cramer's rule over a commutative ring: every solution satisfies det(A)xⱼ=det(Aⱼ(b)), and a unit determinant gives the unique quotient formula Theorem
- Transfer-matrix theorem: weighted-walk generating functions are cofactors of I-xA divided by det(I-xA) Theorem
Dependency tree · two levels
15 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- András Pál, Introduction to Commutative Algebra, Proposition 7.2 (standard reference, not scraped)