Solution: Problem 1

a) The transition matrix A is $\left[ \begin{array}{rr} 0.92&0.04\\ 0.08&0.96\\ \end{array}\right]$

b) The number of math students after one year is the first component of the vector $x_1= A x_0$.

$x_1= A x_0 = \left[ \begin{array}{rr} 0.92&0.04\\ 0.08&0.96\\ \end{array}\ri... ...\\ \end{array}\right] \left[ \begin{array}{r} 198\\ 352\\ \end{array}\right]$

Hence the number of the mathematics major is 198.

c) The number of biology students after three years is 355. It is determined by $X_3=A^3 X_0$

Since $A^3 = \left[ \begin{array}{rr} 0.7876 & 0.1062\\ 0.2124 & 0.8938\\ \end{array}\right]$

$x_3 = A^3 x_0 = \left[ \begin{array}{rr} 0.7876 & 0.1062\\ 0.2124 & 0.8938\... ...ay}\right] = \left[ \begin{array}{r} 194.6912\\ 355.3088\\ \end{array}\right]$

Hence there will be 355 Biology majors at senior year.