I need to prove the following: If matrix A is nonsingular, then the eigenvalues of A inverse are the reciprocals of the eigenvalues of A.

Here is what I know:

det(A-lambdaI)=0

det(A inverse)=1/det(A)

I feel like I need to use this second fact somehow in my proof, but I am struggling with how to get to this step.

Help!