Yes , your derivation is correct
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 ⅄).
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