Question 35533
Given that λ is an eigenvalue of the invertibe matrix with x as its eigen vector.
This means Ax = λx such that x is non-zero

Ax = λx

lets multiply both side of the above equation by the inverse of A( A^-1) from the left. This is possibe since the inverse of A exits according to the problem definition.

(A^-1)*A*x = (A^-1)*λx

since (A^-1)*A  = I(identity matrix)

(A^-1)*λx = x

muliply by 1/λ both sides
(A^-1)*x = 1/λx

this shows that 1/λ is eigenvalue of matrix A^-1 with eigen vector x