Find the eigenvalues from the eigenvalues of a matrix

Question

I don't understand this question how to solve it?

If the eigenvalues of a matrix $A$ are $\lambda_1 = 1, \lambda_2 = 4.79, \lambda_3 = 0.21 $

a) Find the eigenvalues of $A^{-1}$.
b) Find the eigenvalues of $A^2$.
c) Find the eigenvalues of $5A$.

Answer 1

Hints (require some self work):

(1)See what happens when $A$ is diagonal.

(2) If $A$ is diagonalizable, then $A=PDP^{-1}$ when $D$ is diagonal.

Answer 2

HINT: Start with the fundamental eigenvector equation $Ax=\lambda x$. What do you get if you multiply both sides by $A^{-1}$? by $A$? by $5$?