A white rat is put into the maze at time 0. At each instant n = 1, 2, . . ., it moves to a
new compartment, or stays in the same compartment, with probabilities 1/3 and 2/3 respectively. If
it moves, and there are k ways to leave the compartment, it chooses one of these at random. If the rat
starts out in compartment 8, find the expected time at which it returns to compartment 8.

I didn't include the maze's picture, but here is the transition matrix I composed based on the compartments:
*zer is 0 so that they line up evenly
2/3 1/3 zer zer zer zer zer zer zer
1/6 2/3 1/6 zer zer zer zer zer zer
zer 1/6 2/3 zer zer 1/6 zer zer zer
zer zer zer 2/3 zer zer 1/3 zer zer
zer zer zer zer 2/3 zer zer 1/3 zer
zer zer 1/3 zer zer 2/3 zer zer zer
zer zer zer 1/3 zer zer 2/3 zer zer
zer zer zer zer 1/9 zer 1/9 2/3 1/9
zer zer zer zer zer zer zer 1/3 2/3

How do I find the expected value above based on this?