A school has three clubs, and each student is required to belong to exactly one club. One year the students switched club membership as follows:
Club A. 1/5 remain in A, 2/5 switch to B, and 2/5 switch to C.
Club B. 1/4 remain in B, 1/2 switch to A, and 1/4 switch to C.
Club C. 1/6 remain in C, 1/2 switch to A, and 1/3 switch to B.
If the fraction of the student population in each club is unchanged from year to year, find the fraction of the student population in each club every year. Write your answers as common fraction, not decimals.
I don't think there are any special equations required, just an understanding of matrices.
We are supposed to use a graphing calculator (TI-83) to solve the problem, but I can't figure out how to set up the equation.
The Attempt at a Solution
My first attempt was set up as a 3x3 using the remaining students for each club as in/out flow chart, but that didn't go anywhere.
I am currently trying to set up the equations assuming that the student body is A+B+C=1 or (100%).
A B C
|(1/5) (-2/5) (-2/5) (1)|
|(-1/2) (1/4) (-1/4) (1)|
|(-1/2) (-1/3) (1/6) (1)|
|(1) (1) (1) (1)|
I plugged it in to the calculator and it gave me
Back to the drawing board... I just really don't know where to start.