inventory math problem with transition matrix

Project Description:

consider an inventory system in which the sequence of events during each period is as follows:
(1) we observe the inventory level (call it i) at the beginning of the period.
(2) if i <= 1, 4 – i units are ordered.
if i >= 2, 0 units are ordered.
(3) with probability 1/3, 0 units are demanded during the period;
a. with probability 1/3, 1 unit is demanded during the period; and
b. with probability 1/3, 2 units are demanded during the period.
(4) we observe the inventory level at the beginning of the next period.

define a period’s state to be the period’s beginning inventory level. determine the transition matrix that could be used to model this inventory system as a markov chain.

[note: you must provided a fully worked solution and not just answers. i need to see how the answer was derived]
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