Matrix powers (transition / Markov chains)

Find the state after n steps, or the long-run steady state, by raising the transition matrix to a power.

In short: Raising a matrix to a power repeatedly applies the same transformation, which is how transition (Markov chain) matrices predict the state after many steps. Enter the matrix, then raise it to the power n with the power key. Multiply by the starting state vector to read the probabilities after n steps.

When you'd use this

A population or system moves between states with fixed transition probabilities each step.
Finding the distribution after a specific number of steps (raise the transition matrix to that power).
Finding the long-run (steady-state) distribution as the power gets large.
A question explicitly describes a Markov chain or transition matrix.

TI-84 Plus CE Casio fx-CG50 and fx-CG100 TI-Nspire CX

At a glance: TI-84 Plus CE, Casio fx-CG50 and TI-Nspire CX compared

	TI-84 Plus CE	Casio fx-CG50 and fx-CG100	TI-Nspire CX
Key sequence	Enter [A] in 2nd → MATRX → EDIT, then compute [A]^n × [B] on the home screen.	Run-Matrix → MAT to enter the matrix; compute Mat A ^ n × the state vector.	Enter the matrix, then type matrix ^ n × the state vector.

On a TI-84 Plus CE

Enter the transition matrix and the initial state vector.
Enter [A] in 2nd → MATRX → EDIT, then compute [A]^n × [B] on the home screen.
For the long-run state, raise the matrix to a large power (e.g. ^50) and read the stabilising column.

Tip: Columns of a high power of a regular transition matrix converge to the steady-state distribution.

On a Casio fx-CG50 and fx-CG100

Enter the transition matrix and the initial state vector.
Run-Matrix → MAT to enter the matrix; compute Mat A ^ n × the state vector.
For the long-run state, raise the matrix to a large power (e.g. ^50) and read the stabilising column.

Tip: Columns of a high power of a regular transition matrix converge to the steady-state distribution.

On a TI-Nspire CX

Enter the transition matrix and the initial state vector.
Enter the matrix, then type matrix ^ n × the state vector.
For the long-run state, raise the matrix to a large power (e.g. ^50) and read the stabilising column.

Tip: Columns of a high power of a regular transition matrix converge to the steady-state distribution.

Related guides

Try it yourself

Here's a real IB-style question that uses exactly this technique.

Hard Calculator Paper 2 [4 marks]

Each year, 80% of people in region A stay in A and 20% move to B; 30% of people in B move to A and 70% stay in B. Starting with 100 people in A and 0 in B, find the numbers in A and B after 3 years.

A: 65, B: 35

Mark it

Correct 4 / 4 marks

Worked solution & mark scheme:

M1 Set up the transition matrix T = [[0.8, 0.3], [0.2, 0.7]]

M1 Compute T³ × [100, 0]

A1 A ≈ 65, B ≈ 35

Common questions

When would I need to raise a matrix to a power for transition or Markov chain questions in IB Maths?

Find the state after n steps, or the long-run steady state, by raising the transition matrix to a power. A population or system moves between states with fixed transition probabilities each step. Finding the distribution after a specific number of steps (raise the transition matrix to that power).

How do I raise a matrix to a power for transition or Markov chain questions on a TI-84 Plus CE?

1. Enter the transition matrix and the initial state vector. 2. Enter [A] in 2nd → MATRX → EDIT, then compute [A]^n × [B] on the home screen. 3. For the long-run state, raise the matrix to a large power (e.g. ^50) and read the stabilising column.

How do I raise a matrix to a power for transition or Markov chain questions on a Casio fx-CG50 and fx-CG100?

1. Enter the transition matrix and the initial state vector. 2. Run-Matrix → MAT to enter the matrix; compute Mat A ^ n × the state vector. 3. For the long-run state, raise the matrix to a large power (e.g. ^50) and read the stabilising column.

How do I raise a matrix to a power for transition or Markov chain questions on a TI-Nspire CX?

1. Enter the transition matrix and the initial state vector. 2. Enter the matrix, then type matrix ^ n × the state vector. 3. For the long-run state, raise the matrix to a large power (e.g. ^50) and read the stabilising column.

What should I watch out for when I raise a matrix to a power for transition or Markov chain questions?

Columns of a high power of a regular transition matrix converge to the steady-state distribution.

How are marks awarded when I raise a matrix to a power for transition or Markov chain questions in an IB exam?

In the worked example on this page (4 marks, Paper 2), the marks are: M1: Set up the transition matrix T = [[0.8, 0.3], [0.2, 0.7]]; M1: Compute T³ × [100, 0]; A1: A ≈ 65, B ≈ 35.

Practise with your calculator

Questions that need this technique link back to this guide. Try one in practice mode, or see every GDC guide.