A  =  {{(1 + (Sqrt[5]))/2,  (1 - (Sqrt[5]))/2}, {1, 1}}

MatrixForm[A]

MatrixForm[Inverse[A]]

Ainv  =  Inverse[A]

Diag =  {{(1 + (Sqrt[5]))/2,  0}, {0, (1 - (Sqrt[5]))/2}}

MatrixForm[Diag]

Simplify[A . (Diag^n) . Ainv . {1, 1}]