You should know . Now, if is invertible, then exists and . Anyway...
Since we have the product of two matrices (in this case ) equal to a scalar multiple of the second matrix (in this case, ), that means that must be an eigenvalue of .
Suppose A is nonsingular. Prove that the eigenvalues of A^-1 are the reciprocals of the eigenvalues of A.
I was able to come up with a proof for eigenvalues of A being eigenvalues of A^-1, but I'm unsure about how to prove the opposite way to prove the equality.