Exercise 10

Students in a school are sometimes absent due to an illness. Suppose that

• 95% of the students who attend school will attend school the next day.

• 50% of the students who are absent one day will be absent the next day.

We will record the number of present students $p$ and the number of absent students $a$ in a state vector $\xvec=\twovec{p}{a}$ and note that that state vector evolves from one day to the next according to the transition function $T:\real^2\to\real^2$. On Tuesday, the state vector is $\xvec=\ctwovec{1700}{100}$.

1. Suppose we initially have 1000 students who are present and none absent. Find $T\left(\ctwovec{1000}{0}\right)$.

2. Suppose we initially have 1000 students who are absent and none present. Find $T\left(\ctwovec{0}{1000}\right)$.

3. Use the results of parts a and b to find the matrix $A$ that defines the matrix transformation $T$.

4. If $\xvec=\ctwovec{1700}{100}$ on Tuesday, how are the students distributed on Wednesday?

5. How many students were present on Monday?

6. How many students are present on the following Tuesday?

7. What happens to the number of students who are present after a very long time?