Solution: Problem 2

W The transition matrix is the same as the one for problem 1.

Eigenvalues of $A$ are :

$$\lambda_1 = 22/25,\hspace{2cm} \lambda_2 = 1$$

and corresponding eigenvectors are :

${\bf v_1} = \left[ \begin{array}{r} -985/1393\\ 985/1393\\ \end{array}\right]... ...bf v_2}= \left[ \begin{array}{r} -1292/2889\\ -2584/2889\\ \end{array}\right]$

The diagonal matrix D and matrix $P$ are:

$D = \left[ \begin{array}{rr} 22/25&0\\ 0&1\\ \end{array}\right]\hspace{2cm} P... ...n{array}{rrr} -985/1393&-1292/2889\\ 985/1393&-2584/2889\\ \end{array}\right]$

Therefore,

$x_{15} = P D^{15} P^{-1}x_0 = \left[ \begin{array}{rr} -985/1393&-1292/2889 ... ...ray}\right] =\left[ \begin{array}{r} 28239/152\\ 8377/23\\ \end{array}\right]$

There is 186 people have interest in mathematics fifteen years later.

b) Similarly you can compute that there will be 365 people interested in biology twenty years later.