Aⁿ Matrix Power Calculator

Matrix Power Calculator

Matrix power Aⁿ is computed by multiplying the matrix A by itself n times, and this calculator handles any positive or negative integer exponent with each step shown. Supports A⁰ = I, positive powers by repeated multiplication, and negative powers via A⁻¹.

Aⁿ

Matrix Power Aⁿ

Compute matrix raised to any integer power

✓ Any Integer✓ Negative Powers✓ Free⚡ Loading precision engine…
Matrix A
Quick Examples

Matrix Powers

Raising a matrix to a power means multiplying it by itself that many times: A³ = A × A × A. This only works for square matrices (same number of rows and columns), since you can't multiply two non-square matrices by themselves.

$$A^n = \underbrace{A \times A \times \cdots \times A}_{n \text{ times}}$$

How It's Calculated

This calculator multiplies the matrix by itself repeatedly, once for each power. For A⁰, the result is always the identity matrix (a matrix of 1s on the diagonal and 0s elsewhere) — the matrix equivalent of the number 1.

Worked Example (2×2)

For A = [[2,0],[0,3]] and n = 2: A² = A × A = [[4,0],[0,9]]. Each diagonal entry is simply squared, because this particular matrix only has values on its diagonal.

Applications

  • Modeling repeated processes, like population growth over several time periods
  • Computer graphics (repeating a transformation, like a rotation, multiple times)
  • Graph theory (counting paths of a certain length between points)

Frequently Asked Questions

Yes — any square matrix raised to the power of 0 gives the identity matrix (1s on the diagonal, 0s everywhere else), the same way any number raised to the power of 0 equals 1.
A negative power means taking the inverse of the matrix first, then raising that to the positive power. This only works if the matrix has an inverse (its determinant isn't zero).
Multiplying a matrix by itself requires the number of columns in the first copy to match the number of rows in the second. This is only guaranteed when the matrix is square.

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