I've been given the following problem and I want to know if the answer that I found makes sense.
A student center cafeteria has three fast-food centers - serving burgers, tacos, and pizza.A survey of students found the following information concerning lunch: 75% who ate burgers will eat burgers again at the next lunch, 5% will eat tacos next, and 20% will eat pizza next. Of those who ate tacos last, 20% will eat burgers next, 60% will stay with tacos, and 20% will eat pizza next. Of those who ate pizza, 40% will eat burgers next, 20% tacos, and 40% pizza again. Assume initially that one-third of students ate at each of the burger, taco, and pizza stations.
Find the long-term behavior of the students regarding fast food. Explain and interpret your findings. What I found is the following matrix(approximate numbers):
$\begin{pmatrix} .55 & .55 & .55\\ .19 & .19 & .19 \\ .25 & .25 &.25 \end{pmatrix}$
So when I explain my findings I believe that this is a steady state and the matrix has reached equilibrium. So in the long run 55% of the students who ate burgers will eat burgers again the next day, 55% of the students that ate tacos will eat burgers the next day, 55% of the students that ate pizza will eat burgers the next day, so on and so forth. Does this sound accurate??
Not really. On the long run $55\%$ of the students will eat burgers.
The equation to calculate the steady state is
$A\cdot \vec x=\vec x$
Thus your result is a vector not a matrix.
$\vec x=\left(\begin{array}{} \frac59 \\ \frac{7}{36} \\ \frac14 \end{array}\right)$
Your results are approximately right. $\checkmark$