I came across the following problem that says:
Let $A$ and $B$ be $n \times n$ real matrices such that $AB=BA=0$ and $A+B$ is invertible. Then how can I prove the following:
rank $A$+ rank $B$= $n$
nullity $A$ + nullity $B$ =$n$
$A-B$ is invertible.
Can someone point me in the right direction? Thanks in advance for your time.