TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

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Matrix arithmetic over a commutative ring is associative, unital and distributive, and transpose reverses products

Statement

For matrices over a commutative ring, whenever the shapes are compatible, $A(BC)=(AB)C,\qquad I_mA=A=AI_n,$ $A(B+C)=AB+AC,\qquad (A+B)C=AC+BC,$ and $(AB)^{\mathsf T}=B^{\mathsf T}A^{\mathsf T},\qquad (A^{\mathsf T})^{\mathsf T}=A.$ The corresponding entrywise additive and scalar laws also hold, including all zero-sized shapes.

Facts & Assumptions

Given: A commutative ring $R$ and matrices of compatible finite shapes over $R$ .

[L1]

Matrix addition, multiplication, identity and transpose are given by their entrywise finite-sum formulas (Entrywise ring-matrix operations, rectangular matrix products, identity matrices and transpose).

[L2]

Finite sums in a commutative monoid may be reindexed and finite double sums may be taken in either order (Finite commutative-monoid sums are invariant under bijective reindexing, split over disjoint unions, and satisfy the finite Fubini rule).

[L3]

Ring multiplication distributes over addition, annihilates zero and obeys the usual sign laws (In any ring $0 \cdot a = a \cdot 0 = 0$ , $(-a)b = a(-b) = -(ab)$ , $(-a)(-b) = ab$ , $(-1)a = -a$ and $a(b - c) = ab - ac$ ).

Proof

technique · direct

1.1

Entrywise ring addition gives its abelian-group laws. Substitution in the product formula gives $I_mA=A=AI_n$ and both distributive laws, since all off-diagonal identity terms are zero and the diagonal term is one.

L1L3algebra

2.1

For compatible $A,B,C$ , the $(i,\ell)$ entry of $(AB)C$ is $\sum_k(\sum_j a_{ij}b_{jk})c_{k\ell}$ , while that of $A(BC)$ is $\sum_j a_{ij}(\sum_k b_{jk}c_{k\ell})$ . Distributivity and finite-sum Fubini turn both into $\sum_{j,k}a_{ij}b_{jk}c_{k\ell}$ .

step 1.1L2L3algebra

3.1

The $(k,i)$ entry of $(AB)^{\mathsf T}$ is $\sum_j a_{ij}b_{jk}=\sum_j b_{jk}a_{ij}$ , the $(k,i)$ entry of $B^{\mathsf T}A^{\mathsf T}$ ; transposing twice restores each entry. Empty sums are zero, so every argument also covers zero-sized shapes.

step 2.1L1L2algebra∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 47 results over 12 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)