Problem 1179

Is the following transition matrix regular?

A =

 

Solution:-

 

A transition matrix is regular if same power of the matrix contains all positive entries.

The given matrix is A, or A¹. Since 0 is not a positive number, A¹ does not contain all positive entries.

Thus, other powers of a must be examined to determine whether or not A is regular.

Next, look at A², which if found by multiplying A by itself using matrix multiplication.

A² =

 

Notice that both A and A² contain only one zero and in both matrices this zero is in row 1 and column 2. For any transition matrix P, if all zeros occur in the identical places in both Pⁿ and Pⁿ⁺¹ for any n, they will appear in those places for all higher powers of P. Therefore there would be no power of P that contain all positive entries, so P is  not regular.

Thus, the given transition matrix A is nor regular because the only 0 occurs in the identical place in both A¹ and A², which means there is no power of A that contains all positive entries.