t the end of each year the faculty for a graduate program examine the progress that

each graduate student has made on their required thesis. Past records indicate that

$30\%$ of graduate students complete the thesis requirement $($C$)$ and $10\%$ are dropped

from the program for insufficient progress $($D$),$ never to return. The remaining students

continue to work on their theses and remain current graduate students $($G$).$

$($a$)$ Construct a $3\backslash$times $3$ matrix A that describes the student population$$s movement

within the graduate program over each one year period. Label the rows and

columns of A beginning with G and ending with D$.$

$($b$)$ Currently, there are $100$ graduate students in the program and no students have

completed their thesis or been dropped from the program. Find an initial popu$-$

lation vector x$0$ for the graduate student population.

$($c$)$ Using A$,$ predict how many students remain graduate students $($G$),$ have com$-$

pleted their thesis $($C$),$ and have been dropped from the program $($D$)$ one year

from now.

$($d$)$ Using A$,$ predict how many students remain graduate students $($G$),$ have com$-$

pleted their thesis $($C$),$ and have been dropped from the program $($D$)$ two years

from now.

$2$