Markov Transition Matrix

[oswald]

A dynamic interurban of people shows the following Markov Transition Matrix of residents to urban, suburban and rural areas:

__________Urban___Suburban____Rural
Urban ........... a................b...............y
suburban...... o..............q..............z
Rural ............ 1-a-o .........1-b-q ....... 1-y-z

A = 0.9
O = 0.05
B = 0.1
Q = 0.7
Y = 0.1
Z = 0.1

Solve the problem knowing that the vector answer today is
Uo ............ 10
(SUo) = (.... 40)
Ro ............ 50


What vector response to t = 2 vector and what is the answer in the long run. Remember that
Ut
(SUt) = Aλ1 ^ tX1 + Bλ2 ^ tX2 + Cλ3 ^ tX3
Rt

any bibliography suggestion?

[EnumaElish]

Here is one: http://en.wikipedia.org/wiki/Stochastic_matrix

And another: http://mathworld.wolfram.com/StochasticMatrix.html

[oswald]

[q1 q2 q3] x [ 0,1 -0,05 -0,05] = [ 0 0 0 ]
....................-0,7..1......-0,3
....................-0,8..0......0,8

0,1q1 - 0,7q2 -0,8q3 =0
-0,5q1 +q2 = 0
-0,5q1 -0,3q2+0,8q3=0

q1+q2+q3=1

q1=?
q2=?
q3=?
Cramer doesnt work!

[oswald]

answer
http://www.yaronet.com/posts.php?s=108496