Matrix Power Calculator

Work out matrix power instantly with clear inputs, formula shown and shareable results.

About Matrix Power Calculator

Matrix powers are computed by repeated squaring, needing only about log₂ k multiplications. The Fibonacci matrix [[1,1],[1,0]] raised to k has Fibonacci numbers as its entries.

Formulas

Determinant of a power

det(Aᵏ) = (det A)ᵏ

Fibonacci matrix

[[1,1],[1,0]]ᵏ = [[F(k+1), F(k)], [F(k), F(k−1)]]

Worked example

With the default values below, Matrix Power Calculator returns Aᵏ of [89, 55] [55, 34]. Change any input above and every figure here is recalculated in your browser.

Inputs used

Square matrix, row by row: 1, 1, 1, 0
Power kaccepts 0–60: 10

Results

Aᵏ: [89, 55] [55, 34]
Top-left entry: 89.000000
Trace of Aᵏ: 123.000000
det(Aᵏ) = (det A)ᵏ: 1.000000

Step by step

1Values used
2Determinant of a power
3Fibonacci matrix
4Aᵏ= [89, 55] [55, 34]
5Top-left entry= 89.000000
6Trace of Aᵏ= 123.000000
7det(Aᵏ) = (det A)ᵏ= 1.000000

Assumptions and limits

The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 6 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

What is [[1,1],[1,0]]¹⁰?

[[89, 55], [55, 34]] — the Fibonacci numbers F₁₁, F₁₀ and F₉.

Why is A⁰ the identity?

By the same convention as x⁰ = 1: the empty product of matrices is the multiplicative identity.