Linear and multilinear algebra; matrix theory Proposition

Distributivity of matrix multiplication

Builds on Matrix multiplication Matrix addition

Statement

Let $R$ be a ring. Let $A$ be an $m \times n$ matrix over $R$ , and let $B$ and $C$ be $n \times p$ matrices over $R$ . Then $A(B + C) = AB + AC$ . Let $D$ and $E$ be $m \times n$ matrices over $R$ , and let $G$ be an $n \times p$ matrix over $R$ . Then $(D + E)G = DG + EG$ .

Remarks

Both identities follow entry by entry from the distributive law in $R$ . We need both identities, because matrix multiplication is not commutative. Together with associativity, they make the square matrices a ring.

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Matrix definition
Matrix addition definition
Matrix multiplication definition
Distributivity of matrix multiplication statement

Tag	`0RXD6EZYBK`
URI	`m2://0RXD6EZYBK`
Kind	statement, proposition
Writing	W0 stub
MSC	15A99
Status	This copy of the site has no build data. Run `m2 status 0RXD6EZYBK`.
Uses	Matrix multiplication, Matrix addition

Distributivity of matrix multiplication

Statement

Remarks

Reading path

Further reading

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