Chapter 1.6, Problem 44E

Let $A,B,$ and $C$ be $\left(m\times n\right)$ matrices such that $A+C=B+C$. The following statements are steps in a proof that $A=B$. Using Theorem 7, provide justification for each of the assertions.

a) There exists an $\left(m\times n\right)$ matrix $O$ such that $A=A+O$.

b) There exists an $\left(m\times n\right)$ matrix $D$ such that $A=A+\left(C+D\right)$.

c) $A=A+\left(C+D\right)=\left(B+C\right)+D$.

d) $A=B+\left(C+D\right)$.

e) $A=B+O$.

f) $A=B$