Is the following transition matrix regular?
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.
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.