Question

A study about residence and housing conducted by a university revealed that 80% of students living on campus opt to remain on campus the following year. also, 30% of students who live off campus choose to move to on-campus the following year.

1. if there were x students living on campus and y students living off campus in 2015-2016, calculate how many live on-campus in housing in 2016-2017 (using matrix multiplication).
2. Are there values of x and y so that there is no change in the on-campus and off-campus numbers year- over-year? Give some examples. What proportion of the student population must x be?
3. Relate your results to the eigenvalues and eigenvectors of the matrix you used in (1).

Answer 1

To calculate the number of students living on-campus in 2016-2017, we can use matrix multiplication. The proportion of the student population that x represents is x / (x + y).

Step-by-step explanation:

To calculate the number of students living on-campus in 2016-2017, we can use matrix multiplication. Let x represent the number of students living on campus in 2015-2016, and y represent the number of students living off campus in 2015-2016.

The matrix representing the transition from on-campus to on-campus is [0.8, 0.3], and the matrix representing the transition from off-campus to on-campus is [0.2, 0.7]. So, the matrix representing the total transition is [0.8, 0.3] * [x, y].

In order to find the number of students living on-campus in 2016-2017, we can multiply the matrices: [0.8, 0.3] * [x, y] = [0.8x + 0.3y, 0.8x + 0.3y]. This means that the number of students living on-campus in 2016-2017 is 0.8x + 0.3y.

To calculate the proportion of the student population that x represents, we can divide x by the total number of students: x / (x + y).