If you compute the eigenvalue decomposition of

  A={{0,1},{1,1}}

i.e.

  A = V^-1 D V

where D is a diagonal matrix with the eigenvalues phi and 1/phi as the diagonals, then

  A^n = V^-1 D^n V

where D^n is diagonal with phi^n and 1/phi^n as the diagonals. This gives exactly the above well known formula.