> You can exponentiate the matrix n-1 times to get a matrix that converts F_1 and F_0 (known in advance) into F_n and F_{n-1}. This is just as amenable to the shortcuts involved in calculating large exponents of a value as exponentiating ϕ is. But it's exact integer arithmetic.

Right, and repeated squaring can give you the answer to the exponentiation in logarithmic time.