Solving a system of 4 equations

**mybrohshi5** · March 6th 2012, 04:09 PM

I am trying to solve a problem for my stochastic process class and I am running into some problems solving the system of equations I have obtained from my transition matrix. It has been a while since I have taken algebra, or any class dealing with solving systems of equations, so please forgive me for my easy question

I am trying to solve $\pi * P = \pi$ where $\pi = (\pi_0, \pi_1, \pi_2, \pi_3)$ and $P = \left( \begin{array}{cccc} 0.7&0.3&0&0 \\ 0&0&0.4&0.6 \\ 0.5&0.5&0&0 \\ 0&0&0.2&0.8 \end{array} \right)$

$\pi * P = (.7\pi_0+.5\pi_2,.3\pi_0+.5\pi_2,.4\pi_1+.2\pi_3,. 6\pi_1+.8\pi_3) = (\pi_0, \pi_1, \pi_2, \pi_3)$

So I basically need help solving this system of equations:

$.7\pi_0+.5\pi_2= \pi_0$
$.3\pi_0+.5\pi_2 = \pi_1$
$.4\pi_1+.2\pi_3 = \pi_2$
$.6\pi_1+.8\pi_3 = \pi_3$

Also note: $\pi_0 + \pi_1 + \pi_2 + \pi_3 = 1$

Thank you in advance

**Wilmer** · March 6th 2012, 07:47 PM

Originally Posted by mybrohshi5

$.7\pi_0+.5\pi_2= \pi_0$
$.3\pi_0+.5\pi_2 = \pi_1$
$.4\pi_1+.2\pi_3 = \pi_2$
$.6\pi_1+.8\pi_3 = \pi_3$

"Unmess(!)" your equations by changing variables to a,b,c,d and multiplying by 10:
7a + 5c = 10a
3a + 5c = 10b
4b + 2d = 10c
6b + 8d = 10d

Now play with that "silly" system

**mybrohshi5** · March 7th 2012, 06:31 AM

Thank you Wilmer. That made it much easier to solve

Answer should be a=1/4, b=c=3/20, and d=9/20

To check: a+b+c+d = 1

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