DefinitionDefinition: AI-adaptedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-13

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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.
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For $A \in M_{n} (R)$ , the coordinate endomorphism $T_{A} : R^{n} \to R^{n}$ , with $\det (T_{A}) : = \det (A)$ and $adj (T_{A}) : = T_{adj (A)}$

Definition

Let $R$ be a commutative ring and $n \geq 1$ . Write $R^{n} : = M_{n \times 1} (R)$ , with entrywise addition and scalar multiplication (Finite rectangular matrices over a commutative ring, their entries, rows and columns). For $A \in M_{n} (R)$ , define the coordinate endomorphism

$T_{A} : R^{n} \to R^{n}, T_{A} (x) : = A x,$

using matrix multiplication (Entrywise ring-matrix operations, rectangular matrix products, identity matrices and transpose). It preserves addition and scalar multiplication. Define

$\det (T_{A}) : = \det (A), adj (T_{A}) : = T_{adj (A)},$

where the right sides use the matrix determinant (For $n \geq 1$ , the determinant over a commutative ring by the Leibniz formula, and $∣ \det A ∣$ for a real matrix) and the matrix adjugate (Deleted-row-and-column minors, cofactors, the cofactor matrix and the adjugate over a commutative ring). These definitions concern the specified coordinate self-map; no choice of basis is involved.

Depends on

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

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 58 results over 14 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

András Pál, Introduction to Commutative Algebra (standard reference, not scraped)

For A∈Mn(R)A∈Mn​(R), the coordinate endomorphism TA:Rn→RnTA​:Rn→Rn, with det⁡(TA):=det⁡(A)det(TA​):=det(A) and adj⁡(TA):=Tadj⁡(A)adj(TA​):=Tadj(A)​