Problem 1180

Find the equilibrium vector for the given matrix.

P = $\begin{bmatrix} 0.3 & 0.7 \\ 0.4 & 0.6 \end{bmatrix}$

Solution:-

First, we should determine if the transition matrix P is regular.

Since all of the entries are positive, P is regular.

Since it is regular, we can find a probability vector V where VP = V.

V will be the equilibrium vector. Let V = [x, y]. Then

VP = [x y] $\begin{bmatrix} 0.3 & 0.7 \\ 0.4 & 0.6 \end{bmatrix}$

= [0.3x + 0.4y 0.7x+0.6y]

Set the entries equal to each other

0.3x + 0.4y = x and 0.7x +0.6y = y

Simplify both equations.

-0.7x + 0.4y = 0 and 0.7x – 0.4y = 0

Therefore are really the same equation, so we need a second equation.

Since V is a probability vector, x + y = 1.

Now solve the system.

-0.7x + 0.4y = 0

x + y = 1

[x y ] = [ $\frac{4}{11} \frac{7}{11}$ ]