At 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 $(G).$

$2$

$($a$)$ Construct a $33$ 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 population vector $x_0$ for the graduate student population.

$($c$)$ Using $A,$ predict how many students remain graduate students $($G$),$ have completed 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 completed their thesis $($C$),$ and have been dropped from the program $($D$)$ two years from now.