What Is a Transition Matrix and How Does It Model a Discrete Dynamical System?

A transition matrix records how quantities or probabilities move between states during each time step. It models a discrete dynamical system by repeatedly multiplying the current state vector by the matrix.

The Mechanism

In IB Mathematics: Applications and Interpretation HL, AHL 4.19, the usual convention is that the entry $T_{ij}$ gives the probability of moving from state $j$ to state $i$. Therefore, each column of $T$ must sum to $1$.

The model is

$$\mathbf{s}_n=T^n\mathbf{s}_0.$$

Suppose customers choose companies A or B each month. Of company A's customers, $80\%$ remain with A, while $20\%$ move to B. Of company B's customers, $30\%$ move to A and $70\%$ remain with B.

$$T=\begin{pmatrix}0.8&0.3\\0.2&0.7\end{pmatrix},\qquad \mathbf{s}_0=\begin{pmatrix}700\\300\end{pmatrix}.$$

After one month,

$$\mathbf{s}_1=T\mathbf{s}_0 =\begin{pmatrix}650\\350\end{pmatrix}.$$

After two months,

$$\mathbf{s}_2=T^2\mathbf{s}_0 =\begin{pmatrix}625\\375\end{pmatrix}.$$

Both state vectors still total $1000$, so multiplication redistributes customers rather than creating or removing them. The first component, $650$, comes from $0.8(700)+0.3(300)$, while the second comes from $0.2(700)+0.7(300)$. This shows how each new state combines contributions from every previous state.

For a probability state vector, the entries sum to $1$. For a population vector, they sum to the conserved total population, provided no customers enter or leave the model.

Thus, the system changes only at separate monthly steps. It is a Markov chain because the next state depends only on the current state, not on the system's earlier history. If repeated multiplication approaches a fixed vector, that vector is the steady state, satisfying $T\mathbf{s}=\mathbf{s}$.

A common misconception is to place “from” states in rows. Under the standard IB convention, “from” states are columns, although you must follow any different convention explicitly stated in the question.

Exam Technique

For an AHL 4.19 Paper 1 or Paper 2 question, write $T^n\mathbf{s}_0$ before using your GDC, check that each column sums to $1$, and interpret the resulting entries in context. Keep full calculator precision and round only the final answer.