Let $A$ , $B$ be complex $n \times n$ matrices. Which of the followings are true?
- If $ A$,$B$ and $A+B$ are invertible, then $A^{-1}+B^{-1}$ is invertible.
- If $ A$,$B$ and $A+B$ are invertible, then $A^{-1}-B^{-1}$ is invertible.
- If $AB$ is nilpotent, then $BA$ is nilpotent.
- Characteristic polynomial of $AB$ and $BA$ are equal if $A$ is invertible.
Clearly 1 is true and I found a example in 2 that 2 is not correct, but I have no idea on 3 and 4. Kindly help me.