= 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 $(1/2,0,1/2)^{\mathsf T}$. Hence
$$
\boxed{A=\frac14,\qquad B=0,\qquad C=\frac12,\qquad P=\begin{pmatrix}1/2&1/2&1/2\\1/4&1/2&0\\1/4&0&1/2\end{pmatrix}.}
$$
Each column sums to one, as required for this convention for a <stochastic matrix>. Multiplying by $P$ conserves the total resources.
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