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.