Matrix powers describe repeated transformations: k steps of a Markov chain, the k-th Fibonacci number ([[1, 1], [1, 0]]ᵏ) or k applications of a rotation.
How to use the Matrix Power Calculator
- Choose the matrix size – the matching grid of fields appears.
- Enter the entries row by row (a₁ b₁ c₁ is the first row, a₂ b₂ c₂ the second, and so on).
- Read the result; integer matrices give exact fractions.
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
Squaring and multiplying (binary exponentiation) needs only about log₂ k multiplications. For negative k the inverse is raised to |k|.
Formula
Aᵏ = A · A · … · A (k times), A⁰ = I, A⁻ᵏ = (A⁻¹)ᵏ
Worked example
Inputs — Matrix size: 2 × 2; a₁: 1; b₁: 1; a₂: 1; b₂: 0; Power k: 10.
Results — Aᵏ = ⎡ 89 55 ⎤
⎣ 55 34 ⎦; det(Aᵏ) = det(A)ᵏ = 1.
Key terms
- Matrix power
- Repeated multiplication of a square matrix by itself.
Frequently asked questions
What is A⁰?
The identity matrix.
How do I compute A¹⁰⁰ efficiently?
Repeated squaring: A², A⁴, A⁸ … then combine.
What is [[1, 1], [1, 0]]ⁿ?
[[F(n+1), F(n)], [F(n), F(n−1)]] – Fibonacci numbers.
Can I raise a non-square matrix to a power?
No.