Past exam of the year 1 of the computer science course of the University of Oxford / 2013 / Digital systems, linear algebra, introduction to formal proof / 6 / e / Solution

The factor $(\lambda -1)$ proves that one is an eigenvalue. To find its normalized eigenvector, solve $Pv=v$. The second and third rows giveso $v_2=v_3=v_1/2$. The normalization $v_1+v_2+v_3=1$ now givesIts entries are nonnegative and sum to one, and direct matrix multiplication verifies $Pv=v$. As a probability vector it is the stationary distribution; as resource shares it assigns half to the USA and a quarter to each other region.