This limit does not exist (the sequence diverges).

As long as [1,2,3]' contains a component in the direction of an eigen vector corresponding to the largest absolute value eigen value and that is greater than 1 this will diverge with absolute value eventually growing by a factor equal to the largest absolute value of the eigen values. The iterates will eventually rotate so that they point closer and closer to the direction of the eigen vector in question (after adjusting for the sign anyway).

In this case the eigen value of largest absolute value is

CB