Did you actually calculate the characteristic equation for this matrix? It is . That is the same as saying that and so A^3= -2A^2. There is not that much difference between a cube and a square so this is tedious but doable.

1000/3= 333 and 1/3 so . Now 334/3= 111 and 1/3 so .

So far we have that [tex]A^{1000}= 2^{444}A^{223}. 223/3= 74 and 1/3 so : .

159/3= 53 so . .

106/3= 35 and 1/3 so . .

Keep doing that until you have reduced this to a number times .