Transition matrices and state vectors | Year 12

01 / LEARN

Understand the idea.

States and transitions

A state vector records the amounts in each category at one time. Here it is a column vector and $x_{n + 1} = T x_{n}$ . Rows of $T$ identify next states; columns identify current states, so $t_{i j}$ is the fraction moving from $j$ to $i$ .

This convention differs from row-start adjacency matrices. In a closed transition model, every member must have a destination: each column sums to 1 and each entry lies between 0 and 1. These checks ensure the total is preserved.

Proportions act on current amounts

To forecast A, multiply each current amount by its fraction moving to A, then add. This is the first row-column product. Repeat for every destination; the resulting vector lists predicted amounts, not transition percentages.

Fractional counts represent expected amounts. Keep full precision during calculations. If the population gains or loses members, model the extra flow or redefine the states; do not force a deficient column to total 1 without investigating what is missing.

x_{n + 1} = T x_{n}

Columns=current; rows=next.
Next / current	A	B
A	0.8	0.3
B	0.2	0.7

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

A one-period forecast

Apply the displayed matrix to the state (100,50).

Worked solution

1
Align current states with matrix columns.
$T = (\begin{array}{cccccccccc} 0.8 & 0.3 \\ 0.2 & 0.7 \end{array}), x = (\begin{array}{cccccccccc} 100 \\ 50 \end{array})$
Both use A, B order; the first column gives destinations for current A and the second for current B.
2
Calculate next-period A.
$A_{next} = 0.8 (100) + 0.3 (50) = 80 + 15 = 95$
Add those remaining in A to those moving from B into A.
3
Calculate next-period B.
$B_{next} = 0.2 (100) + 0.7 (50) = 20 + 35 = 55$
Add those moving from A into B to those remaining in B.
4
Check and report the state.
$T x = (\begin{array}{cccccccccc} 95 \\ 55 \end{array}), 95 + 55 = 100 + 50 = 150$
The total remains 150, as required by the closed-system model.

Answer

State column (95,55).

EXAMPLE 02

Missing transition entries

The first column is (0.65, $p$ ), the second ( $q$ ,0.9). Find $p, q$ .

Worked solution

1
Complete the first current-state column.
$\begin{aligned} 0.65 + p & = 1 \\ p & = 0.35 \end{aligned}$
All destinations for the first current state must account for its entire population.
2
Complete the second current-state column.
$\begin{aligned} q + 0.9 & = 1 \\ q & = 0.10 \end{aligned}$
The second column must also sum to 1.
3
Check the resulting transition matrix.
$T = (\begin{array}{cccccccccc} 0.65 & 0.10 \\ 0.35 & 0.90 \end{array})$
Both columns sum to 1 and all four entries are between 0 and 1.

Answer

$p = 0.35, q = 0.10$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

For rows (0.7,0.2),(0.3,0.8), what fraction goes B to A?

Show answer & reasoning

0.2, or 20%.

Use current-state column B and next-state row A. Their intersection contains 0.2, so 20% of current B moves to A in one period.

Keep practising

6 worksheet questions · 20 marks

Open worksheet Worked solutions

Read the questions online

Use rows $(\begin{array}{cccccccccc} 0.7 & 0.2 \\ 0.3 & 0.8 \end{array})$ to update $(80, 20)$ .
3 marks
Rows of a proposed closed transition matrix are $(0.4, 0.3)$ , $(0.5, 0.7)$ . Is it valid?
3 marks
Find $a, b$ so rows $(\begin{array}{cccccccccc} 0.75 & a \\ b & 0.6 \end{array})$ form a column-stochastic matrix.
3 marks
A state vector is $(120, 80)$ . What total must a closed transition preserve?
3 marks
Each day 90% of North bikes stay North; 40% of South bikes move North.
5 marks
1. (a)
  Write the North,South column-stochastic matrix.
  2 marks
2. (b)
  Predict tomorrow from $(20, 30)$ .
  3 marks
A forecast is $(34.6, 15.4)$ for 50 bikes. Explain its meaning.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Use rows $(\begin{array}{cccccccccc} 0.7 & 0.2 \\ 0.3 & 0.8 \end{array})$ to update $(80, 20)$ .
3 marks
A: $0.7 (80) + 0.2 (20) = 60$ .
B: $0.3 (80) + 0.8 (20) = 40$ .
$(60, 40)$ .
Rows of a proposed closed transition matrix are $(0.4, 0.3)$ , $(0.5, 0.7)$ . Is it valid?
3 marks
It does not account for 10% of those initially in A.
No; the first column totals 0.9.
Find $a, b$ so rows $(\begin{array}{cccccccccc} 0.75 & a \\ b & 0.6 \end{array})$ form a column-stochastic matrix.
3 marks
$0.75 + b = 1$ and $a + 0.6 = 1$ .
$a = 0.4, b = 0.25$ .
A state vector is $(120, 80)$ . What total must a closed transition preserve?
3 marks
The starting entries sum to 200.
Column sums of 1 preserve this total.
200.
Each day 90% of North bikes stay North; 40% of South bikes move North.
5 marks
a) Each column describes destinations from one depot.
b) North: $0.9 (20) + 0.4 (30) = 30$ .
b) South: $0.1 (20) + 0.6 (30) = 20$ .
1. (a)
  Write the North,South column-stochastic matrix.
  2 marks
  Each column describes destinations from one depot.
  Rows $(\begin{array}{cccccccccc} 0.9 & 0.4 \\ 0.1 & 0.6 \end{array})$ .
2. (b)
  Predict tomorrow from $(20, 30)$ .
  3 marks
  North: $0.9 (20) + 0.4 (30) = 30$ .
  South: $0.1 (20) + 0.6 (30) = 20$ .
  $(30, 20)$ .
a) Rows $(\begin{array}{cccccccccc} 0.9 & 0.4 \\ 0.1 & 0.6 \end{array})$ .; b) $(30, 20)$ .
A forecast is $(34.6, 15.4)$ for 50 bikes. Explain its meaning.
3 marks
An operational integer plan should still total 50.
Do not interpret fractional counts literally.
It is an expected distribution; actual whole-bike counts vary.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 2 General Mathematics matrix-models option. This is the offered fifth-topic option, alongside the four compulsory topics; a school-designed Open Topic is an alternative, not prescribed content in this chapter.

Official curriculum source ↗

In this lesson

Understand the idea.

States and transitions

Proportions act on current amounts

See the method in action.

A one-period forecast

Align current states with matrix columns.

Calculate next-period A.

Calculate next-period B.

Check and report the state.

Missing transition entries

Complete the first current-state column.

Complete the second current-state column.

Check the resulting transition matrix.

Try it yourself.

Check your understanding

Keep practising