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Matrix Power Calculator

Compute A^n by repeated matrix multiplication for 2x2 and 3x3 matrices

Enter 2x2 Matrix

Exponent n

Matrix Power Formula

A^0 = I (identity matrix)

A^1 = A

A^2 = A × A

A^n = A × A × ... × A (n times)

(A^n)^m = A^(n×m)

Matrix powers are computed by repeated multiplication. A^n means multiply A by itself n times. For diagonalizable matrices, you can use eigenvalues for faster computation, but repeated multiplication works for any square matrix.

⚠ Matrix power is only defined for square matrices (n×n) and non-negative integer exponents.

What is Matrix Power?

Matrix power A^n is just multiplying A by itself n times. A^0 is the identity matrix, A^1 is A itself, A^2 is A×A, and so on. This is used in Markov chains, recurrence relations, and many other applications.

A^0 = Identity

Any matrix to the 0 power is identity matrix I.

Repeated Mult

A^n = A multiplied n times in sequence.

Square Matrix

Only square matrices can be raised to powers.

Diagonal Matrices

For diagonal D, D^n just powers each diagonal entry.

Teaching Example: A = [[1,2],[3,4]], n=2
1. A^2 = A×A = [[1×1+2×3, 1×2+2×4],[3×1+4×3, 3×2+4×4]]
2. = [[7, 10],[15, 22]]

Applications

Markov Chains Recurrence Relations Graph Theory Computer Science Dynamics

Frequently Asked Questions

What is matrix power?▼

A^n = A multiplied n times. A^2 = A×A, A^3 = A×A×A, etc.

A^0 = I?▼

Yes, any matrix to the 0 power is the identity matrix.

How to compute A^n?▼

Repeated multiplication, or eigenvalues for diagonalizable matrices.

(A^n)^m = A^(nm)?▼

Yes, exponents multiply like regular numbers.

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