###### Exercise 9

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 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 \(\xvec=\twovec{p}{a}\text{.}\) On Tuesday, the state vector is \(\xvec=\ctwovec{1700}{100}\text{.}\) The state vector evolves from one day to the next according to the transition function \(T:\real^2\to\real^2\text{.}\)

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

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

Use the results of parts a and b to find the matrix \(A\) that defines the matrix transformation \(T\text{.}\)

If \(\xvec=\ctwovec{1700}{100}\) on Tuesday, how are the students distributed on Wednesday?

How many students were present on Monday?

How many students are present on the following Tuesday?

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