Matrix Power Calculator

Enter your square matrix (up to 4×4) and a power (exponent) to raise it to — the Matrix Power Calculator multiplies the matrix by itself the specified number of times and returns the resulting output matrix with all element values displayed. Supports positive integer powers for 2×2, 3×3, and 4×4 matrices.

Select the size of your square matrix.

Enter a non-negative integer power to raise the matrix to.

Used for 3×3 and 4×4 matrices only.

Used for 4×4 matrices only.

Results

Result Matrix

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Matrix Size

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Power Applied

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Trace of Result Matrix

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Result [1][1]

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Result [1][2]

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Result [2][1]

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Result [2][2]

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Result [1][3]

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Result [2][3]

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Result [3][1]

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Result [3][2]

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Result [3][3]

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Result [1][4]

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Result [2][4]

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Result [3][4]

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Result [4][1]

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Result [4][2]

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Result [4][3]

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Result [4][4]

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Result Matrix Element Values

Results Table

Frequently Asked Questions

What is a matrix power?

A matrix power means multiplying a square matrix by itself a specified number of times. For example, A² = A × A, and A³ = A × A × A. The matrix must be square (same number of rows and columns) for this operation to be valid.

Why must the matrix be square to raise it to a power?

Matrix multiplication requires the number of columns in the first matrix to equal the number of rows in the second. Since we are multiplying the matrix by itself, this condition is only satisfied when the matrix is square (n×n).

What does raising a matrix to the power of 0 return?

Any square matrix raised to the power of 0 equals the identity matrix of the same size. The identity matrix has 1s on the main diagonal and 0s everywhere else, similar to how any scalar number raised to the power of 0 equals 1.

What does raising a matrix to the power of 1 return?

A matrix raised to the power of 1 returns the original matrix unchanged. This is consistent with how exponentiation works for regular numbers.

How is matrix squaring (power of 2) different from squaring each element?

Raising a matrix to the power of 2 means performing full matrix multiplication (A × A), which involves dot products of rows and columns. This is entirely different from squaring each individual element, which is an element-wise operation. The two operations produce different results in almost all cases.

What is the trace of the result matrix?

The trace of a matrix is the sum of all elements on its main diagonal (top-left to bottom-right). This calculator shows the trace of the resulting powered matrix as a quick summary statistic useful in linear algebra applications.

What matrix sizes does this calculator support?

This calculator supports 2×2, 3×3, and 4×4 square matrices. For each size, you enter the corresponding number of elements and the desired integer power (0 through 10).

Can this calculator handle large powers like A^10?

Yes, this calculator supports powers up to 10. The computation is performed by repeated matrix multiplication. Note that element values can grow very large for higher powers, particularly for matrices with entries greater than 1.

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