Linear Algebra Done Right · Problem 3C.11

Linear Maps — Sheldon Axler

Mathematics — Linear Algebra · matrix-multiplication

Problem

Prove that the distributive property holds for matrix addition and matrix multiplication. In other words, suppose

A

B

C

D

E

, and

F

are matrices whose sizes are such that

A(B + C)

and

(D + E)F

make sense. Explain why

AB + AC

and

DF + EF

both make sense and prove that

A(B + C) = AB + AC \quad \text{and} \quad (D + E)F = DF + EF.