Chapter 1.6, Problem 45E

Let $A,B,C,$ and $D$ be matrices such that $AB=D$ and $AC=D$. The following statements are steps in a proof that if $r$ and $s$ are scalars, then $A\left(rB+sC\right)=\left(r+s\right)D$. Use Theorems 8 and 9 to provide reasons for each of the steps.

a) $A\left(rB+sC\right)=A\left(rB\right)+A\left(sC\right)$

b) $A\left(rB+sC\right)=r\left(AB\right)+s\left(AC\right)=rD+sD$

c) $A\left(rB+sC\right)=\left(r+s\right)D$