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 / a / Solution

With column vectors, entry $P_{ij}$ is the fraction moving from region $j$ to region $i$. The first column is therefore $(1/2,1/4,1/4{)}^{\mathsf{T}}$, the second is $(1/2,1/2,0{)}^{\mathsf{T}}$, and the third is $(1/2,0,1/2{)}^{\mathsf{T}}$. HenceEach column sums to one, as required for this convention for a stochastic matrix. Multiplying by $P$ conserves the total resources.