I need help with a few matrix problems. Any help is greatly appreciated!?

Question

22. Suppose that
A=[.2 .8]
----[.1 .9]
is a transition matrix.
What is the probability that, that state 1 changes to state 2?

24. Find the absorbing states for the following transition matrix
---1 2 3----
1[1 0 0 ]
2[.2 .3 .5 ]
3[ 0 0 1 ]

25. In the following game, decide on the payoff when the strategy (3,1) is used.
-------1 2 3
---1[ 4 -2 1]
A 2[-3 0 2 ]
---3[-6 3 0 ]

27. Suppose that a game has a payoff matrix:
Suppose that a player A chooses row 1 and with probability 0.3 and player B chooses column with probability 0.5. Find the expected value of the game.

How can you tell if a matrix is a probability vector?
How do you identify a transition matrix?
How do you know if a transition matrix is regular or not???

Answer 1

22. 0.8 - just read this off the matrix.

24. You can see that state 1 will always remain at state 1, and state 3 will always remain at state 3, while state 2 can move to any of the three states. So states 1 and 3 are the absorbing states.

25. When A chooses 3 and B chooses 1, the payoff is -6 (just read this off from the matrix). I'm not entirely sure that this is the correct interpretation of the question - normally "strategy" is used to indicate the probability with which one player will choose each of the alternatives. This is clearly not how it's being used here, so I assume it does mean that A chooses 3 and B chooses 1.

27. Can't answer this without the payoff matrix.

A probability vector has only non-negative entries, which add to 1.
A transition matrix has only non-negative entries, and the entries in each row or in each column add to 1 (depends on whether you're using row or column vectors). It's most usual to use column probability vectors and require the columns of the matrix to add to 1, but the example in Q22 seems to be going the other way.
A transition matrix is regular, by definition, if some power of it has only strictly positive entries.

Answer 2

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