# Matrix Power Calculator

URL: https://speedycalculator.com/calculators/math/matrix-power/
Last updated: 2026-10-30

> The matrix power calculator raises a 2 × 2, 3 × 3 or 4 × 4 matrix to an integer power k (negative powers use the inverse), using fast exponentiation.

## Inputs

- Matrix size
- a₁
- b₁
- c₁
- d₁
- a₂
- b₂
- c₂
- d₂
- a₃
- b₃
- c₃
- d₃
- a₄
- b₄
- c₄
- d₄
- Power k

## Results

- Aᵏ
- det(Aᵏ) = det(A)ᵏ

## Formula

```
Aᵏ = A · A · … · A (k times),  A⁰ = I,  A⁻ᵏ = (A⁻¹)ᵏ
```

## Explanation

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|.

## 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.

## FAQ

**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.


## Sources

- [Wolfram MathWorld – Matrix power](https://mathworld.wolfram.com/MatrixPower.html)
- [MIT OpenCourseWare – 18.06 Linear Algebra (Gilbert Strang)](https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/)
- [Jim Hefferon – Linear Algebra (free textbook)](https://hefferon.net/linearalgebra/)