# Science:Math Exam Resources/Courses/MATH152/April 2022/Question B2 (c)/Solution 1

Suppose that, in the long run, are the proportions of buyers who buy Ambrosia, Braeburn and Cameo apples, respectively. Then the transition matrix should leave the vector fixed. In other words, we are looking for an eigenvector with eigenvalue 1, or an element of the null-space of the matrix

We can find the elements of the null-space by row-reducing. Again, we indicate the row operations as in B1 (a), but we also use for the operation "swap rows and ".

We can easily see from the row-reduced matrix above that eigenvectors of with eigenvalue 1 are of the form If we want the entries of this vector to be proportions of people buying the three different kinds of apples, then the entries of the eigenvector must sum up to 1. The equation implies that , so we have that the long-run shares of each apple variety are