In a large metropolitan area, 20% of the commuters currently use the public transportation system, whereas the remaining 80% commute via automobile. The city has recently revitalized and expanded its public transportation system. It is expected that 6 months from now 40% of those who are now commuting to work via automobile will switch to public transportation, and 60% will continue to commute via automobile. At the same time, it is expected that 20% of those now using public transportation will commute via automobile and 80% will continue to use public transportation. (a) Construct the transition matrix for the Markov chain that describe:s the change in the mode of transportation used by these commuters (b) Find the initial distribution vector for this Markov chain. (c) What percentage of the commuters are expected to use public transportation 6 months from now? (Round your answer to the nearest percent.)

Answer: [tex]\\[/tex](a) [tex]\left[\begin{array}{ccc}&40&60\\&20&80\\\end{array}\right][/tex]  [tex]\\[/tex] (b) [tex]D_{0}[/tex] = [20      80]   [tex]\\[/tex] (c) 50%Step-by-step explanation:[tex]\\[/tex](a) Since we are constructing the transition matrix for the Markov chain that describes the change in the mode of transportation used by the commuters , we must take note that the matrix must be a square matrix and the row must sum up to be 1, if it is probability and 100 if it is percentage.[tex]\\[/tex]40% switch from  automobile to public, this will be the firs element on the matrix , which is [tex]a_{11}[/tex] , while 60% continues with automobile, this will be [tex]a_{12}[/tex].[tex]\\[/tex]Also, 20% of those now using public transport will commute via automobile , this will be [tex]a_{21}[/tex] and the 80% that continued with public will be [tex]a_{22}[/tex] . Therefore the matrix will be[tex]\\[/tex][tex]\left[\begin{array}{ccc}&40&60\\&20&80\\\end{array}\right][/tex][tex]\\[/tex](b) Initially , it was given that 20% currently use public transport and 80% use automobile , so the initial distribution vector implies [tex]\\[/tex][tex]D_{0}[/tex] = [20      80][tex]\\[/tex](c) Those that currently use public transport = 20% [tex]\\[/tex]Those that currently use automobile = 80% [tex]\\[/tex]6 month from now, 40% 0f 80 will switch to public transport, that is [tex]\\[/tex]40/100 x 80 = 32 [tex]\\[/tex]That means the remaining automobile = 80 – 32     [tex]\\[/tex]= 48 [tex]\\[/tex]Also, 20% of 20 will switch to automobile, that is [tex]\\[/tex]20/100 x 20 = 4 [tex]\\[/tex]The remaining public transport = 20 – 4 [tex]\\[/tex]= 16 [tex]\\[/tex]Therefore, in 6 months’ time, the total number of those that will use public transport will be [tex]\\[/tex]32 + 16 = 48% [tex]\\[/tex]To the nearest % = 50%