Check fixed-population and regular transition models practice questions | General Mathematics (Year 12, VCE Units 3 & 4, Transition matrices, Transition matrices and diagrams)

Validate a transition model

A transition matrix uses columns for current states and rows for next states. Validate the model conditions first, then test regularity using a positive power rather than inspecting only T.

S_{n+1}=T S_n,\qquad \sum_{i=1}^{n}t_{ij}=1\quad\text{and}\quad t_{ij}\ge 0

Read columns as current states Column j gives the proportions moving from current state j to each next state \(\displaystyle \sum_{i=1}^{3}t_{i1}=\frac{1}{2}+\frac{1}{2}+0=1\)
Check non-negative entries Every transition proportion must be at least zero \(\displaystyle t_{ij}\ge 0\)
Use the current state only A one-step model must determine the next state from the current state, not from an earlier state \(\displaystyle S_{n+1}=T S_n\)
Test a positive power A matrix is regular when some positive power has every entry strictly greater than zero \(\displaystyle T=\begin{pmatrix}\frac{1}{2}&\frac{1}{2}&0\\ \frac{1}{2}&0&\frac{1}{2}\\ 0&\frac{1}{2}&\frac{1}{2}\end{pmatrix},\quad T^2=\begin{pmatrix}\frac{1}{2}&\frac{1}{4}&\frac{1}{4}\\ \frac{1}{4}&\frac{1}{2}&\frac{1}{4}\\ \frac{1}{4}&\frac{1}{4}&\frac{1}{2}\end{pmatrix}>0\)

Worked example

A proposed transition model has three states in the order A, B, C. Each column represents a current state and each row represents a next state. Decide whether it is a fixed-population one-step model and whether it is regular

T=\begin{pmatrix}\frac{1}{2}&\frac{1}{2}&0\\ \frac{1}{2}&0&\frac{1}{2}\\ 0&\frac{1}{2}&\frac{1}{2}\end{pmatrix}

\(\displaystyle \frac{1}{2}+\frac{1}{2}+0=1,\quad \frac{1}{2}+0+\frac{1}{2}=1,\quad 0+\frac{1}{2}+\frac{1}{2}=1\)All entries are non-negative and the three current-state columns each total 1, so the matrix conditions hold
\(\displaystyle \left(T^2\right)_{3,1}=0\times\frac{1}{2}+\frac{1}{2}\times\frac{1}{2}+\frac{1}{2}\times 0=\frac{1}{4}\)The entry in row 3, column 1 is found by multiplying row 3 of T by column 1 of T.
\(\displaystyle T^2=\begin{pmatrix}\frac{1}{2}&\frac{1}{4}&\frac{1}{4}\\ \frac{1}{4}&\frac{1}{2}&\frac{1}{4}\\ \frac{1}{4}&\frac{1}{4}&\frac{1}{2}\end{pmatrix}\)Every entry of this second power is strictly positive
\(\displaystyle \left(T^2\right)_{ij}>0\text{ for all }1\le i,j\le 3\quad\Rightarrow\quad T\text{ is regular}\)The model is valid, and a positive power with every entry strictly positive establishes regularity

Common mistakes

Rows instead of columns

✕\(\displaystyle \sum_{j=1}^{n}t_{ij}=1\)✓\(\displaystyle \sum_{i=1}^{n}t_{ij}=1\)

Each column represents one current state, so its entries must total 1; a row total is not the required fixed-population check

Rejecting a zero in T

✕\(\displaystyle t_{ij}=0\;\Rightarrow\;T\text{ is not regular}\)✓\(\displaystyle t_{ij}=0\;\not\Rightarrow\;T\text{ is not regular};\quad T^2>0\text{ can establish regularity}\)

Regularity requires some positive power to have every entry strictly positive, so a zero in T may disappear in a later power

Using reachability alone

✕\(\displaystyle \text{Every state can reach every other state}\;\Rightarrow\;T\text{ is regular}\)✓\(\displaystyle \text{Some }k\ge 1\text{ has }(T^k)_{ij}>0\text{ for all }i,j\;\Rightarrow\;T\text{ is regular}\)

Mutual reachability does not rule out periodic zero entries, so a strictly positive power is still required

Difficulty levels

introCheck the square matrix, non-negative entries and current-state column totals

standardDistinguish a regular model from an absorbing or reducible model using a displayed power

challengeSeparate periodic strongly connected models from regular models and recognise the current-state-only assumption

examChoose the statement that correctly combines the fixed-population and regularity conclusions

Questions about this topic

Why must the columns of a transition matrix total 1?+

Each column gives the proportions leaving one current state. The proportions must account for the whole population, so they are non-negative and sum to 1.

Can a transition matrix with a zero entry still be regular?+

Yes. Regularity depends on some positive power of the matrix having every entry strictly positive. A zero in T can disappear when transitions are combined

Why is reachability alone not enough to prove regularity?+

A strongly connected cycle can return to a state only at selected step counts, leaving zeros in every individual power. Regularity needs one positive power with no zero entries

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 12, VCE Units 3 & 4, General Mathematics, chapter 11: Transition matrices

In this chapter

11A Transition matrices and diagrams: Translate transition descriptions, diagrams and matrices 11A Transition matrices and diagrams: Check fixed-population and regular transition models 11B Interpreting transition matrices: Translate transition descriptions, diagrams and matrices Use a transition matrix to predict the next step 11C State recurrences and equilibrium: Check fixed-population and regular transition models Generate state matrices by recursion 11C State recurrences and equilibrium: Apply and reverse a state recurrence with external change Determine the long-run state of a transition model Construct a transition model with a specified equilibrium 11D Adjusted state recurrences: Apply and reverse a state recurrence with external change Construct and interpret Leslie matrices and life-cycle diagrams Generate and total Leslie population states Analyse long-run behaviour of a Leslie model