Find the transition matrix for a six-sided die

Question

For some reason I have been struggling with this problem for the past couple hours.

I believe I have solved part a.

Since there are 6 states (assuming a standard die and the die is fair), then there is an equal chance of landing in any of the 6 states for every dice throw. Therefore I concluded that the transition matrix looks as such:

[1 0 0 0 0 0] [0 1 0 0 0 0] [0 0 1 0 0 0] [0 0 0 1 0 0] [0 0 0 0 1 0] [0 0 0 0 0 1]

For part b I setup my starting matrix as:

s1 =

[1] [0] [0] [0] [0] [0]

I multiplied my transition matrix by s1 5 times and arrived at the same vector s1. Arithmetically, this makes sense. However, logically it does not.

Can somebody help explain where I went wrong?

Edit: Based on the answers I believe the transition matrix would be:

[1/6 0 0 0 0 0] [1/6 2/6 0 0 0 0] [1/6 1/6 3/6 0 0 0] [1/6 1/6 1/6 4/6 0 0] [1/6 1/6 1/6 1/6 5/6 0] [1/6 1/6 1/6 1/6 1/6 1]

Answer 1

Hint. The entry $m_{i,j}$ in the transition matrix is the probability to go to state $i$ at the next throw given that you are currently in state $j$. Therefore it is lower triangular, but not quite the identity.

E.g. if you are in state $4$ you have a $4/6$ chance to stay in state $4$. Therefore $m_{4,4}=4/6$.

Answer 2

From state $s_1$ you have $1/6$ chance of remaining in that state and a $1/6$ chance of transitioning to any of the other states.

From state $s_2$ you have a $2/6$ chance of remaining there (by getting either a $1$ or a $2$) anad a $1/6$ chance of transitioning to each of $3,4,5,6.$

From state $s_3$ you have a $3/6$ chance of remaining there (by getting $1,$ $2,$ or $3$) and a $1/6$ chance of transitioning to each of $4,5,6.$

From state $s_4$ you have a $4/6$ chance of remaining there (by getting $1,$ $2,$ $3,$ or $4$) and a $1/6$ chance of transitioning to each of $5,6.$

From state $s_5$ you have a $5/6$ chance of remaining there (by getting $1,$ $2,$ $3,$ $4,$ or $5$) and a $1/6$ chance of transitioning to $6.$

From state $s_6$ you can only remain there.