Let $A,B$ be $n \times n$ matrices and $AB=BA$. Show that $$\text{rank}(A+B) \leq \text{rank}(A) + \text{rank}(B) – \text{rank}(AB)$$
Attempt at the solution:
I first proved that $$R(A+B) \subseteq R(A) + R(B)$$ where R(A) is Range(A).
Then I used the dimension formula that states that $$\dim(U+V) = \dim(U) + \dim(V) – \dim(U \cap V)$$ so that I get $$r(A+B) \leq r(A) +r(B) – \dim(R(A)\cap R(B))$$
The last part would be to show that $$\dim(R(A)\cap R(B)) = \dim(R(AB))$$ using the fact that $AB=BA$.
I need help with this last part. Also please critique me if the above solution is acceptable or if there is something wrong with it.
Thank you!!
Best Answer
You do not need $\dim(R(A)\cap R(B)) = \dim(R(AB))$. All you need is $\dim(R(A)\cap R(B)) \ge \dim(R(AB))$, which is easy to prove using $AB=BA$.