Ok so for this question I'm having trouble understanding how the transition graph has been drawn from the given transition matrix. This is what I understand and hopefully someone can correct the flaws in my understanding.
Each entry in the transition matrix represents a probability. Column 1 is state 1, column 2 is state 2 and so on up to column 6 which is state 6. Now starting from the first entry in the matrix with value 1/2, we go from state 1 to state 2 with p=1/2. Then from state 2 to state 3 with p=1 then from 3 to state 1 again with p=1/3. Then I don't understand what the loop means but I think it means we start the cycle again and end up back at state 3. So now from state 3 we go to state 4 with p=1/3. Then we should go from state 4 to state 5 with p=1/3 and then state 5 back to state 4 with p=1/2 then from state 4 to state 5 with p=1/2 and the from state 5 to state 6 with p=1 and then from state 6 back to state 5 with p=1.
Now the way I'm looking at this is by looking at each entry and seeing which state it goes to in the same row, obviously I'm going wrong somewhere and don't understand this process properly. Also don't understand the loops on 1 and 4.
Any help would be much appreciated.
The first entry in the matrix is $p_{1,1}=1/2$. It means that with $50$% probability you stay in state $1$ for one time step.
To the right of $p_{1,1}$ you have $p_{1,2}$ which is the probability that you go from state $1$ to state $2$.
Each entry $p_{i,j}$ means that you go from state $i$ to state $j$ where the $i$ is the row and the $j$ is the column.
"Column 1 is state 1, column 2 is state 2 and so on up to column 6 which is state 6."
Row $1$ is when you go from $1$ and column $1$ is when you go to $1$.
"Now starting from the first entry in the matrix with value 1/2, we go from state 1 to state 2 with p=1/2."
No the first entry is $p_{1,1}$. The probability that you stay in state $1$. The "second entry" is $p_{1,2}$, and this is where you go from state $1$ to state $2$.
"Then from state 2 to state 3 with p=1 then from 3 to state 1 again with p=1/3. "
Yes, $p_{2,3} = 1$ and $p_{3,1} = 1/3$.
"Then I don't understand what the loop means but I think it means we start the cycle again and end up back at state 3. "
No. This i have already mentioned several times. The loop is $p_{1,1}$.
"So now from state 3 we go to state 4 with p=1/3."
Yes $p_{3,4} = 1/3$ but also $p_{3,5} = 1/3$ so you can go to state $5$ from state $3$ too.
"Then we should go from state 4 to state 5 with p=1/3."
No, in the fourth row you have two entries $p_{4,4} = 1/2$, which again is a loop and the probability that you stay in state $4$ for one time step. The second entry in row $4$ is $p_{4,5}=1/2$. Which mean you go to state $5$ with probability $1/2$, i.e. when you are in state $4$ you either stay there or go to state $5$, with equal probability.
"and then state 5 back to state 4 with p=1/2"
No, in row 5 you only have one entry $p_{5,6}=1$, which means that with certainty you go to state $6$ when you are in state $5$.
"then from state 6 back to state 5 with p=1."
Yes, only one entry in row $6$, $p_{6,5}=1$. You go to state $5$ with certainty.