How do I solve the following problem step by step? and what are the solutions? I

Question

How do I solve the following problem step by step? and what are the solutions? I am really lost, and do not understand the whole exercise..

The city of Mawtookit maintains a constant population 300,000 people from year to year. A political science study estimated that there were 150,000 Independents, 90,000 Democrates, and 60,000 Republicans in the town. It was also estimated that each year 20 percent of the Independents become Democrats and 10 persent become Republicans- Similarly, 20 persent of the Democrats become Independents and 10 persent become Republicans, while 10 persent of the Republicans defect to the Democrates and 10 persent become Independents each year. Let

and let x(1) be a vector representing the number of people in each group after 1 year.

a) Find a matrix A such that Ax=x(1)

b) Show that 1=1.0 2=0.5 and 3=0.7 are eigenvalues of A, and factor A into a product XDX-1, where D is diagonal.

Explanation / Answer

Let A and B be similar matrices. Since they are similar, there exists a matrix P such that B =P-1 A P. Now,by eigenvalues of B, Det(B-In)=Det(P-1 A P-In)=Det(P-1 A P-?In)=Det(P-1 A P- P-1(?In) P)= Det(P-1( A P- P-1(?In) P))=Det(P-1 (A -(?In)) P)=Det(P-1)*Det(A-?In)*Det(P)=Det(P-1)*Det(P)*Det(A-?In)=Det(P-1P)Det(A-?In))=Det(In)*Det(A-?In)=Det(A-?In). Therefore, Det(B-In)=Det(A-?In) and so A and B have the same characteristic equation, and so have the same eigenvalues. QED For the second part, in general similar matrices will have different eigenvectors. If you find any two similar matrices you will surely find your counterexample. (Have to leave some work to you! :) )

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