If $A,B\in M_n(\mathbb{R})$ such that $AB=BA=O$ and $A+B$ is invertible ,then select the correct statements $\ldots$
$(a)$ rank(A)=rank(B);
$(b)$ rank(A)+rank(B)=$n$;
$(c)$ nullity(A)+nullity(B)=$n$;
$(d)$ $A-B$ is invertible.
I've proved $(d)$ is true. As If $A$ is a $n \times n$ matrix ,then it goes a vector space with $n$ dimension to an another vector space and similarily for $B$ but when $BA=0$ it means that every base from image of $A$ will be at kernel of $B$. So $B$ at least has all of basis of image $A$, therefore null $B \geq$ rank $A$.
For $(a),(b),(c)$ I took some examples and found $(a)$ is not correct. But I'm unable to prove $(b),(c)$. Please help me to prove these.
