Originally Posted by

**arbolis** The problem has certainly already be solved but I didn't find it in a short research.

Anyway I'm just asking for a check-result.

Prove that if $\displaystyle A$ and $\displaystyle B$ are $\displaystyle n\times m$ matrices and $\displaystyle C$ is a $\displaystyle m\times q$ one, then $\displaystyle (A+B)C=AC+BC$.

My attempt : By definition, $\displaystyle (A+B)_{ij}=A_{ij}+B_{ij}$ so $\displaystyle (A+B)_{ij}C_{ij}=\sum_{k=1}^n [A_{ik}+B_{ik}]C_{kj}$ (call this 1)$\displaystyle =\sum_{k=1}^n A_{ik}C_{kj}+B_{ik}C_{kj}$ (call this 2)$\displaystyle =AC+BC$.

I'm not sure if I can pass from 1 to 2 as I did or I need more calculus/or explanation.