Suppose that we have a nucloetide string consisting of A, C, G, and T. We are interested in the regions which are "CG rich". In one of the "CG rich" regions of a few thousand letters of DNA we have that the chance that an A will follow an A is .18, the chance that an A will follow a C is .274, the chance that an A will follow a G is .426 and the chance that an A will follow a T is .12. Also, we have that the chance that a C will follow an A is .17, the chance that a C will follow a C is .368, the chance that a C will follow a G is .274 and the chance that a C will follow a T is .188. Further, we have that the chance a G will follow an A is .161, the chance that a G will follow a C is .339, the chance that a G will follow a G is .385 and the chance that a G will follow a T is .115. Finally, we have that the chance a T will follow an A is .079, the chance that a T will follow a C is .355, the chance that a T will follow a G is .384 and the chance that a T will follow a T is .182.

a.) From the information above, show the 1-step stochastic matrix for moving from 1 letter in the code above to the next letter. What are the eigenvalues and corresponding eigenvectors of this matrix?

b.) Using the information in a, state whether a steady-state (stationary) vector exists for the code. If one does exist, find it and interpret it in the context of the propoortion of A's, C's, G's, and T's we expect to find further along the code. If no steady-state vector exists, say whether we'll see a significant rise in the proportion of C's and the proportion of G's, or whether we will see a loss (that is, an extinction) of C's & G's further along the code.