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.
Entrywise ring-matrix operations, rectangular matrix products, identity matrices and transpose
Definition
Let be a commutative ring. Matrices of one shape are added and scaled entrywise. For and , their product is with The sum is the finite sum in the additive commutative monoid of , and is when .
The identity matrix has entry on the diagonal and elsewhere. The transpose is defined by . The zero matrix has every entry .
Depends on
Used by
- The determinant is alternating and multilinear in the rows as well as in the columns Corollary
- Invertible square matrices and similarity over a commutative ring Definition
- For n≥ 1, determinant is a natural transformation det:GLₙ(-)⟹(-)^× from commutative rings to groups Example
- FALSE: det(A+B)=det(A)+det(B) for all same-sized square matrices False statement
- For a field, the ring-matrix operations, invertibility and similarity agree exactly with the established field-matrix interface Proposition
- For every square matrix over a commutative ring, det(A^mathsf T)=det(A) Theorem
- Matrix arithmetic over a commutative ring is associative, unital and distributive, and transpose reverses products Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 51 results over 17 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- S. New, MATH 146 Linear Algebra 1 Lecture Notes, Ch. 4 (standard reference, not scraped)
- P. Massot, Structures algébriques fondamentales, §6.4 (standard reference, not scraped)