Yeah, I'm trying to remember what I know about raising a matrix to a power. Oh, the natural number powers of a matrix of natural numbers are again natural numbered matrices. So we can then reinterpret them as a graph. Which means given a pointed graph that generates a sequence, we can construct a graph that generates every nth value of the series. Which is pretty non obvious I think!
Well the determinant of the product is the product of the determinants, so \\(1 \centerdot 0 - 1 \centerdot 1 = -1\\) means that for the Fibonacci graph, the determinant of the power is going to alternate from negative one and one.