Stationary matrix given a transition matrix

Question

I am given the following transition matrix

$$P= \pmatrix{ 1-\alpha & \alpha \\ \beta & 1-\beta}, \ \alpha,\beta \in (0,1)$$

with the states $S=\{1,2\}$.

I want to determine the stationary distribution $\pi$ of the Markov chain determined by a starting distribution and the transition matrix $P$.

We have the system:

$\pi(1)=(1-\alpha)\cdot \pi(1) + \beta\cdot\pi(2)$

$\pi(2)=\alpha\cdot \pi(1) + (1-\beta)\cdot\pi(2)$

When I substitute for $\pi(2)$ from the first equation in the second, I get $0=0$. However, the answer should be:

$$\pi(1)=\frac{\beta}{\alpha+\beta},\pi(2)=\frac{\alpha}{\alpha+\beta}$$.

What am I doing wrong? Thank you for your time.

Answer 1

You are missing $\pi_1 + \pi_2 = 1$ giving you the proper second equation.