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Multinational companies in the Americas, Asia, and Europe have assets of $\ 4$ trillion. At the start, $\ 2$ trillion are in the Americas and $\ 2$ trillion are in Europe. Each year $1 / 2$ of the Americas money stays home and $1 / 4$ goes to each of Asia and Europe. For Asia and Europe, $1 / 2$ stays home and $1 / 2$ is sent to the Americas.
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a) Let $C_{k}$ be the column vector with the assets of the Americas, Asia, and Europe at the beginning of year $k$. Find the transition matrix $T$ that gives the amount in year $k+1: C_{k+1}=T C_{k}$
b) Find the eigenvalues and eigenvectors of $T$.
c) Find the limiting distribution of the $\ 4$ trillion as the world ends
d) Find the distribution of the $\ 4$ trillion at year $k$.
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