Textbook Problems T16 and T17
I am having a hard time proving questions T16 and T17 from section SD. I am not sure where to start. Does anyone have any ideas or can explain what I should do? I am putting the questions themselves below.
T16: Suppose that A and B are similar matrices, with A being nonsingular. Prove that B is nonsingular, and that $A^{-1}$ is similar to $B^{-1}$.
T17: Suppose that B is a nonsingular matrix. Prove that AB is similar to BA.
Comments
I will give you a couple of hints.
T16
Write down what it means to say that $A$ is similar to $B$. Don't forget that the product of nonsingular matrices is again nonsingular.
T17
Since $B$ is nonsingular, you can try to conjugate $AB$ by $B$.
Thank you so much. These hints really helped a lot!