Aperiodicity of a markov chain

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my transition matrix is

0 0 1
0 0 1
(1/3) (2/3) 0

I'm supposed to argue that this chain is aperiodic,

A markov chain is aperiodic iff there exists a time n such that there is a positive probability of going from state i to state j for all i and j

This doesn't seem to hold for my chain ... for example, to go from state 1 to state 2 n has to be odd.. but to go from state 1 to state 1 or 3 n has to be even..

Am I just getting this definition muddled up? Could someone elaborate on it for me? Thanks
 

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