Consider the matrix P in Example 6(b) 0.5 O o o O 0 0.4 0.5 0 0 0.6 0.5 Is it po
ID: 3117287 • Letter: C
Question
Consider the matrix P in Example 6(b) 0.5 O o o O 0 0.4 0.5 0 0 0.6 0.5 Is it possible to find a steady state matrix X for the corresponding Markov chain? If not, explain why. O Yes, it is possible to find a steady state matrix for the corresponding Markov chain. No, it is not possible as the Markov chain is ot absorbing If so, find a steady state matrix. (If the steady state matrix does not exist then enter DNE. If the system has an infinite number of solutions, express x1, x2, x3, and x4 in terms of the parameter t.) DNEExplanation / Answer
The steay state vector, if it exists, is the solution of the equation PX = X or, (P-I4)X = 0. To solve this equation, we will reduce to its RREF, as ubder, the matrix P-I4=
-0.5
0
0
0
0.5
0
0
0
0
0
-0.6
0.5
0
0
0.6
-0.5
Multiply the 1st row by -2
Add -1/2 times the 1st row to the 2nd row
Interchange the 2nd row and the 3rd row
Multiply the 2nd row by -5/3
Add -3/5 times the 2nd row to the 4th row
Then the RREF of P-I4 is
1
0
0
0
0
0
1
-5/6
0
0
0
0
0
0
0
0
Now, if X = (x1,x2,x3, x4)T, then the equation (P-I4)X = 0 is equivalent to x = 0,and z-5w/6 = 0 or, z = 5w/6. Then X = (0,y,5w/6,w)T= y(0,1,0,0)T+w/6(0,0,5,6)T= r(0,1,0,0)T+t(0,0,5,6)Twhere r,t are arbitrary real numbers. Thus, the steady state vector does not exist.
DNE.
-0.5
0
0
0
0.5
0
0
0
0
0
-0.6
0.5
0
0
0.6
-0.5