Question: suppose that A is an invertible matrix (i.e. A^(-1) ), and that x is an eigenvector for A with eigenvalue ⅄ ≠ 0. Show that x is an eigenvector for A^(-1) with the eigenvalue ⅄^(-1) (inverse ⅄).

My answer:

By: Ax = b, therefore: x = A^(-1) b

When a matrix A and a nonzero vector x satisfy : Ax = ⅄x (for some scalar ⅄), and by: x = A^(-1) b,

then: x = (A^(-1)) ⅄x,

therefore: (⅄^(-1))x = (A^(-1))x

Am i right? or at least on the right track?

please and thankyou