In a city, two written newspapers, "El Ayer" and "El Maana", compete to attract readers. The total number of subscriptions between the two newspapers is always stable and reaches 150,000 subscribers. A marketing study reveals that every year that passes, 10% of the subscribers of "El Ayer" change their subscription to the newspaper "El Maana". Therefore, 90% of the subscribers of the newspaper "El Ayer" continue with their subscription the following year. Instead, 20% of the subscribers to the newspaper "El Maana" will switch to the newspaper "El Ayer" the following year. Consequently, the remaining 80% will continue to be subscribers to the newspaper "El Maana". These estimates can be expressed by a transition matrix given by:

T=(0.90.10.20.8)

We will denote by ( ,) the vector that represents the number of subscribers to the newspaper "El Ayer" (A) and the newspaper "El Maana" (M), respectively, in the year. So, if we know the number of subscribers to "El Ayer" and "El Maana" in year, we can know the respective number of subscribers in year + 1, through the transition matrix by performing the following operation :

(Mt+1At+1)=T(MtAt)

Suppose that in the first year of observation, year = 0, the number of subscribers to "El Ayer" is 120,000 subscribers, while those of the newspaper "El Maana" are 30,000 subscribers:

(M0A0)=(30.000120.000)

With these data it is requested:

a) What are the eigenvalues and the eigenvectors of the transition matrix?

b) What are the subscribers of each newspaper expected to be in the long term? (Note: if and are the diagonal and change of base matrices, respectively, of the matrix, and given that = ^-1, first calculate lim ^t and then lim (MtAt))