Question

Find the​ steady-state vector for the matrix below: {0.7, 0.2, 0.2}, {0.1, 0.6, 0.1}, {0.2, 0.2, 0.7}

The numbers listed here are the rows of a 3x3 matrix.

Answer 1

The given matrix is A =

0.7	0.2	0.2
0.1	0.6	0.1
0.2	0.2	0.7

If X = (x,y,z)^T is the steady state vector, then AX = X or, (A-I₃)X = 0. To solve this equation, we need to reduce the matrix A-I₃ to its RREF which is

1	0	-1
0	1	-0.5
0	0	0

Hence, the equation (A-I₃)X = 0 is equivalent to x-z = 0 or, x = z and y-0.5z = 0 or, y = z/2. Then, X =(z,z/2,z)^T =(z/2)(2,1,2)^T.

Hence, (2,1,2)^T is the steady state vector for the given matrix A.