Prove rank (A + B + AB) ≤ rank (A) + rank (B)

Question

I need to prove so using column and row spaces of the matrix. I don't understand how to express the rank using column space. Is it correct to state that $C(AB) \subseteq C(A)$ since we can write every column of $AB$ as a linear combination of the columns of $A$? If so, how can I connect this fact to the rank of a matrix?

Answer 1

Note that $$A+B+AB=A+(I+A)B.$$

When you combine this with the formulae $$\operatorname{rank}(T+S) \leq \operatorname{rank}(T)+\operatorname{rank}(S) \\ \operatorname{rank}(TS) \leq \min(\operatorname{rank}(T), \operatorname{rank}(S)),$$ we have $$\operatorname{rank}(A+(I+A)B) \leq \operatorname{rank}(A) + \operatorname{rank}((I+A)B) \leq \operatorname{rank}(A) + \operatorname{rank}(B)$$