Simple Markov Chains Memoryless Property Question I have a sequential data from time T1 to T6. The rows contain the sequence of states for 50 customers. There are only 3 states in my data. For example, it looks like this:

        T1  T2  T3  T4  T5  T6
Cust1   C   B   C   A   A   C


My transition matrix X looks like this:

    A   B   C
A   0.3 0.6 0.1
B   0.5 0.2 0.3
C   0.7 0.1 0.2


Now, we see that at time T6 the state is at C which corresponds to c=[0 0 1] vector.
I am now predicting T7 by doing the matrix multiplication: c * X which gives me [0.7 0.1 0.2]. Based on this, I decide that the state at T7 would be A (highest prob. value).

For T8, I use the result of the probability vector I got above and do [0.7 0.1 0.2]*X = [0.4 0.44 0.14]  and decide that the state is B.

My question is: Am I doing something wrong? Am I contradicting the memoryless property of the Markov Chains?
I think you are not doing anything wrong, the markov property is satisfied when the prediction can be solely based on the present state. I do not think you are contradicting this "memoryless" condition (which applies to the whole chain) when you predict what could probaly happen after a last state that is already just predicted.