Question 1:

Let be by real matrices.

If and are invertible, show that is also invertible.

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

Question 2:

Let be by real matrices such that .

Suppose is an eigenvalue of or .

Show that is also eigenvalue of .

Question 3:

Let be by real matrices such that .

If is diagonalizable, show that is diagonalizable.

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

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