For the transition matrix $T = \begin{pmatrix} 0.6 & 0.15 \\ 0.4 & 0.85 \end{pmatrix}$, find the exact steady state probability vector $\mathbf{s} = \begin{pmatrix} x \\ y \end{pmatrix}$.

1. State the steady state condition: multiplying the transition matrix by the steady state vector yields the exact same vector ($T\mathbf{s} = \mathbf{s}$). 2. Set up the matrix equation: $\begin{pmatrix} 0.6 & 0.15 \\ 0.4 & 0.85 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} x \\ y \end{pmatrix}$. 3. Extract the top row into a linear equation: $0.6x + 0.15y = x$. 4. Simplify : $0.15y = 0.4x \implies 15y = 40x \implies 3y = 8x$. 5. Apply the probabilistic constraint that the states must sum to $1$: $x + y = 1 \implies y = 1 - x$. 6. Substitute and solve: $3(1 - x) = 8x \implies 3 - 3x = 8x \implies 11x = 3 \implies x = \frac{3}{11}$. 7. Consequently, $y = \frac{8}{11}$. The exact steady state vector is $\begin{pmatrix} 3/11 \\ 8/11 \end{pmatrix}$ .

Markov Chains and Transition Matrices Questions

What you need to know

Weather forecasting, consumer brand-switching, disease spread. HL AI's Markov questions ask for the state after n steps and the steady-state distribution: the eigenvector of the transition matrix for eigenvalue 1. Markov chains and transition matrices overview →

What's examined in AI HL Markov chains and transition matrices

The question bank covers these Markov chains and transition matrices question types (number of questions in brackets):

Steady State Analysis (7)
Future State Probabilities (5)
Constructing Transition Matrices (1)

Key formulas

Transition matrix (Markov chain): ${\vec{s}}_{n + 1} = T {\vec{s}}_{n}$

In the same notation as the IB formula booklet. All AI HL formulas →

Where students lose marks on Markov chains and transition matrices

Examiner's noteMarkov chains: columns, the initial state, the right power of T

Where marks are lost: Candidates built transition matrices whose columns did not sum to 1 or whose entries were inconsistent, did not know the initial state vector, used the wrong power of T (counting days wrongly), multiplied two powers of T when a two-stage path was needed, left a steady state as a matrix instead of a vector, and used a transition matrix where simple probability was enough. "To which percentage" was read as a change in percentage points.

Examiners flagged this in 4 exam series (May 2024, November 2024, May 2025, November 2025). Summarised in our own words from IB subject reports, IB Programme Resource Centre. Read the full note →

Markov Chains and Transition Matrices worked examples

Worked example 1: Constructing a Transition Matrix · easy

The transition between two states, A and B, is represented by a directed network. The probability of moving from A to B is $0.4$ , and A to A is $0.6$ . The probability of moving from B to A is $0.15$ , and B to B is $0.85$ . Write down the $2 \times 2$ transition matrix $T$ for this Markov chain, ensuring the states are mapped alphabetically.

Solution

4 steps · 1 to fill in

Examiner tip: In the IB AI syllabus, transition matrices are set up so that the columns sum to

1

. If you set it up such that the rows sum to

1

, you will get incorrect state vectors upon multiplication.

Worked example 2: Finding the State Vector after $n$ Transitions · medium

Using the transition matrix $T = (\begin{matrix} 0.6 & 0.15 \\ 0.4 & 0.85 \end{matrix})$ , assume the system initially starts in state A with $100 %$ certainty. Find the probability that the system is in state B after exactly 2 transitions.

Solution

6 steps

Examiner tip: You can simply look at the first column of

T^{2}

to find the probabilities after 2 steps originating entirely from State A. The matrix power

T^{n}

contains all paths of length

n

.

Worked example 3: Calculating Steady State Probabilities · hard

For the transition matrix $T = (\begin{matrix} 0.6 & 0.15 \\ 0.4 & 0.85 \end{matrix})$ , find the exact steady state probability vector $s = (\begin{matrix} x \\ y \end{matrix})$ .

Solution

7 steps · 1 to fill in

Examiner tip: If you try to solve the system using both rows of the matrix equation, you will get infinite solutions because the equations are linearly dependent. You must always swap out one row for the sum constraint

x + y = 1

.

Try these IB Maths AI HL Markov chains and transition matrices questions

Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area. One of them has its full worked solution here, open to everyone.

Question 1 · easy · 4 marks · Paper 1

A system is modelled by the transition matrix $T = (\begin{matrix} 0.7 & 0.2 \\ 0.3 & 0.8 \end{matrix})$ .
The initial state of the system is given by the column vector $s_{0} = (\begin{matrix} 100 \\ 50 \end{matrix})$ .
Calculate the exact state vector after one transition, $s_{1}$ .

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Question 2 · medium · 5 marks · Paper 1

A car rental company operates in two cities, City X and City Y. - A car rented in City X has a $70 %$ chance of being returned to City X, and a $30 %$ chance of being returned to City Y. - A car rented in City Y has a $90 %$ chance of being returned to City Y, and a $10 %$ chance of being returned to City X.
Currently, there are $400$ cars in City X and $200$ cars in City Y.
Using your GDC, find the number of cars expected to be in City X after 5 rental cycles (transitions). Round your answer to the nearest whole car.

Full worked solution Free, no sign-in

Transition matrix (columns = from X, Y): $T = (\begin{matrix} 0.7 & 0.1 \\ 0.3 & 0.9 \end{matrix})$ . [A1]
Initial state $s_{0} = (\begin{matrix} 400 \\ 200 \end{matrix})$ . [A1]
We need $s_{5} = T^{5} s_{0}$ . [M1]
Using GDC: ${(\begin{matrix} 0.7 & 0.1 \\ 0.3 & 0.9 \end{matrix})}^{5} (\begin{matrix} 400 \\ 200 \end{matrix}) = (\begin{matrix} 169.44 \\ 430.56 \end{matrix})$ . [M1]
Number of cars in City X $\approx 169$ cars. [A1]

Every question here has a full worked solution. See them all →

Question 3 · hard · 7 marks · Paper 1

Consider the transition matrix $T = (\begin{matrix} 0.8 & 0.4 \\ 0.2 & 0.6 \end{matrix})$ .
In linear algebra, a scalar $λ$ is an eigenvalue of a matrix $T$ if $det (T - λ I) = 0$ , where $I$ is the identity matrix.
Prove algebraically that $λ = 1$ is an eigenvalue of $T$ , and find the other eigenvalue.

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All 13 Markov chains and transition matrices questions with mark schemes →

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FAQ

How many IB Maths AI HL Markov chains and transition matrices questions are there?

There are 13 exam-style Markov chains and transition matrices questions in the AI HL question bank (Paper 1: 13), graded 4 easy, 4 medium, 5 hard. Every question has a full IB-style mark scheme (M, A and R marks).

Is Markov chains and transition matrices on Paper 1 or Paper 2?

In the question bank these questions are set as Paper 1 questions.

Where can I get the mark schemes?

Open the AI HL Unit 4 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.

IB Maths AI HL Markov Chains and Transition Matrices Questions

Try it first

What you need to know

What's examined in AI HL Markov chains and transition matrices

Key formulas

Where students lose marks on Markov chains and transition matrices

Markov Chains and Transition Matrices worked examples

Worked example 1: Constructing a Transition Matrix · easy

Solution

Worked example 2: Finding the State Vector after $n$ Transitions · medium

Solution

Worked example 3: Calculating Steady State Probabilities · hard

Solution

Try these IB Maths AI HL Markov chains and transition matrices questions

Question 1 · easy · 4 marks · Paper 1

Question 2 · medium · 5 marks · Paper 1

Full worked solution Free, no sign-in

Question 3 · hard · 7 marks · Paper 1

Revise Markov chains and transition matrices for the exam

FAQ

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More AI HL Unit 4 topics

IB Maths AI HL Markov Chains and Transition Matrices Questions

Try it first

What you need to know

What's examined in AI HL Markov chains and transition matrices

Key formulas

Where students lose marks on Markov chains and transition matrices

Markov Chains and Transition Matrices worked examples

Worked example 1: Constructing a Transition Matrix · easy

Solution

Worked example 2: Finding the State Vector after n Transitions · medium

Solution

Worked example 3: Calculating Steady State Probabilities · hard

Solution

Try these IB Maths AI HL Markov chains and transition matrices questions

Question 1 · easy · 4 marks · Paper 1

Question 2 · medium · 5 marks · Paper 1

Full worked solution Free, no sign-in

Question 3 · hard · 7 marks · Paper 1

Revise Markov chains and transition matrices for the exam

FAQ

What else is free here

More AI HL Unit 4 topics

Worked example 2: Finding the State Vector after $n$ Transitions · medium