DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

How 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.

Submatrices and minors of a rectangular matrix

Definition

Let $A = (a_{i j}) \in M_{m \times n} (R)$ be a matrix over a commutative ring (Finite rectangular matrices over a commutative ring, their entries, rows and columns). For increasing lists of distinct row indices $I = (i_{0}, \dots, i_{r - 1})$ and column indices $J = (j_{0}, \dots, j_{s - 1})$ , the submatrix $A [I, J]$ is the $r \times s$ matrix whose $(p, q)$ entry is $a_{i_{p} j_{q}}$ .

When $r = s \geq 1$ , the $(I, J)$ -minor is $\det A [I, J]$ (For $n \geq 1$ , the determinant over a commutative ring by the Leibniz formula, and $∣ \det A ∣$ for a real matrix), and it is called an $r$ -rowed minor. The positive-size condition is part of the terminology here: no determinant of an empty matrix is introduced.

Submatrices and minors of a rectangular matrix

Definition

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