Find the steadystate vector for the transition matrix 6 4 0

Find the steady-state vector for the transition matrix. [.6 .4 0 .1 .8 .1 0 .6 .4]

Solution

Let the given transition matrix be denoted by A. Then, the steady state vector X is the solution to the equation AX = X or, (A-I₃) X = 0. To solve this equation, we will reduce the matrix A-I₃ to its RREF as under:

Multiply the 1st row by -5/2

Add -2/5 times the 1st row to the 2nd row

Multiply the 2nd row by -10

Add -1/10 times the 2nd row to the 3rd row

Add 1/4 times the 2nd row to the 1st row

Then, the RREF of A-I₃ is

1

0

-3/2

0

1

-6

0

0

0

Now, if X = (x,y,z)^T, the equation (A-I₃) X = 0 is equivalent to x -3z/2 = 0 or, x = 3z/2 and y -6z = 0 or, y = 6z. Then X = (3z/2,6z,z)^T = z(3/2,6,1)^T. Hence, the steady state vector is (3/2,6,1)^T.

1	0	-3/2
0	1	-6
0	0	0