Question 1:

Let $\displaystyle A, B$ be $\displaystyle n$ by $\displaystyle n$ real matrices.

If $\displaystyle A, B$ and $\displaystyle A+B$ are invertible, show that $\displaystyle A^{-1} + B^{-1}$ is also invertible.

I tried this question by using elementary matrices. But I still cannot complete the proof.

Question 2:

Let $\displaystyle A, B$ be $\displaystyle n$ by $\displaystyle n$ real matrices such that $\displaystyle AB=0$.

Suppose $\displaystyle \lambda$ is an eigenvalue of $\displaystyle A$ or $\displaystyle B$.

Show that $\displaystyle \lambda$ is also eigenvalue of $\displaystyle A+B$.

Question 3:

Let $\displaystyle A, B$ be $\displaystyle n$ by $\displaystyle n$ real matrices such that $\displaystyle AB=BA$.

If $\displaystyle A$ is diagonalizable, show that $\displaystyle B$ is diagonalizable.

I have no idea where to begin for question 2 & 3.

Please give me some hints for question 2 and question 3.