Use transition matrices and an initial state vector to predict future states and to find the long-term steady state of a system.

What this dot point is asking

You must build a transition matrix from movement data, predict the state after one or more steps, and find or interpret the long-term steady state.

Setting up a transition matrix

A system moves objects (customers, animals, voters) between a fixed set of categories at constant rates each period. The transition matrix $T$ records these rates. Using the convention $S_{n+1} = T S_n$, the entry in row $i$, column $j$ is the proportion moving from category $j$ into category $i$. Each column therefore sums to 1, since everything currently in a category must go somewhere.

Predicting future states

To advance one week, multiply the current state by $T$. Repeat, or use $S_n = T^n S_0$ to jump ahead.

The long-term steady state

As $n$ grows, the state vector usually settles to a fixed steady state $S$ that satisfies $S = TS$ and stops changing from week to week. You can find it by computing high powers $T^n S_0$ until the numbers stabilise, or by solving $S = TS$ with the constraint that the categories add to the total.

Interpreting the result in context

The steady state predicts the eventual market share regardless of where shoppers started, provided the transition rates stay constant. Examiners reward a sentence interpreting the numbers, for example "store B eventually holds about two-thirds of the market."