(Note: You will be permitted no more than 4 attempts for credit on this problem.

Question

(Note: You will be permitted no more than 4 attempts for credit on this problem.) For each of the following transition matrices, determine whether the Markov chain with that transition matrix is regular: (1) Is the Markov chain whose transition matrix whose transition matrix is | 1001 0. 30 0.7 0.8 0. 20 regular? (Yes or No) yes (2) Is the Markov chain whose transition matrix whose transition matrix is ?o 0.5 0.5] |100 [100] regular? (Yes or No) yes (3) Is the Markov chain whose transition matrix whose transition matrix is [001] O 01 0.3 0.1 0.6 regular? (Yes or No) no (4) Is the Markov chain whose transition matrix whose transition matrix is [ 0 1 01 O 0 1 [ 0. 70 0.3] regular? (Yes or No) no (5) Is the Markov chain whose transition matrix whose transition matrix is [O 101 0. 40 0.6 1 0 0 regular? (Yes or No) no

Explanation / Answer

(1) This matrix is not regular since no matter, how many times we multiply the matrix by itself, the first row will remain
1 0 0
as: a2 =
1.0000 0 0
0.8600 0.1400 0
0.8600 0 0.1400?
a3 =  1.0000 0 0
0.9020 0 0.0980
0.9720 0.0280 0
and so on.

(2) a =

0 0.5000 0.5000
1.0000 0 0
1.0000 0 0
a2= 1.0000 0 0
0 0.5000 0.5000
0 0.5000 0.5000
a3= 0 0.5000 0.5000
1.0000 0 0
1.0000 0 0

since a3=a and it contains some zeroes, therefore this matrix is not regular.

(3)

a =

0 0 1.0000
0 0 1.0000
0.3000 0.1000 0.6000
a2=
0.3000 0.1000 0.6000
0.3000 0.1000 0.6000
0.1800 0.0600 0.7600

since it has all positive entries, this matrix is regular.

(4)

a =

0 1.0000 0
0 0 1.0000
0.7000 0 0.3000

a4 =   0.2100 0.7000 0.0900
0.0630 0.2100 0.7270
0.5089 0.0630 0.4281

Hence, It is also a regular markov transition matrix.

I can only solve 4 subparts.

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