Math
Matrix Power Calculator
Enter a square matrix and an integer exponent k to compute A^k exactly.
Positive powers show every intermediate product up to A^6 (larger ones use exponentiation by squaring), A^0 returns the identity, and negative powers use the inverse, with a clear error when the matrix is singular.
One row per line. Must be square. Try [1 1; 1 0] to generate Fibonacci numbers.
Any whole number from -30 to 30. Negative values use the inverse; 0 gives the identity matrix.
Try an example
Result
A^k
—
- Exponent
- —
- det(A)
- —
- det(A^k)
- —
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What a matrix power is
A^k means multiplying the square A rectangular grid of numbers arranged in rows and columns. A by itself k times. A^2 = A·A, A^3 = A·A·A, and so on. Only square matrices can be multiplied by themselves, because the dimension rule needs columns of A to equal rows of A.
How the calculator computes it
- For |k| ≤ 6 it multiplies step by step and shows every intermediate power.
- For larger |k| it uses exponentiation by squaring (A², A⁴, A⁸, …), which needs about 2·log₂(k) multiplications instead of k.
- For k < 0 it first inverts A exactly, then raises the inverse to the power |k|.
- Every entry is an exact fraction, so A^20 of an integer matrix is exact even when the entries have many digits.
How to use it
- Enter a square matrix, one row per line.
- Type the The small raised number saying how many times to multiply the base by itself. k (negative, zero, or positive).
- Read A^k in the panel; open Show the work for the chain of products and the One number computed from a square matrix that tells you whether it can be inverted. check.
Where matrix powers show up
| Situation | Why powers matter |
|---|---|
| Fibonacci numbers | [1 1; 1 0]^k contains F(k+1), F(k), F(k-1) |
| Markov chains | P^k gives the k-step transition probabilities |
| Graph theory | (adjacency matrix)^k counts walks of length k |
| Diagonalization | A = PDP⁻¹ gives A^k = PD^kP⁻¹, with D^k trivial |
Common mistakes
- Squaring each entry instead of multiplying the matrix by itself. A^2 is A·A, not entry-wise squaring.
- Taking a negative power of a Describes a square matrix whose determinant is zero, so it has no inverse. matrix. There is no inverse, so A^(-1) does not exist.
- Expecting fractional powers. A^(1/2) (a matrix square root) is a different, harder problem and is not supported here.
How to type a matrix into this calculator
- Put each row on its own line. Separate the entries in a row with spaces or commas: 2 1 1 or 2, 1, 1.
- Fractions such as 1/3 and decimals such as 0.25 are accepted and kept exact, so 1/3 stays 1/3 instead of becoming 0.3333.
- Every row must have the same number of entries. A row with a missing entry produces a clear error instead of a wrong answer.
- The largest size supported is 8 x 8. Brackets and semicolons are ignored, so [1 2; 3 4] works too.
Worked examples
Fibonacci matrix to the 5th
[1 1; 1 0]^5 = [8 5; 5 3]
A^k
[8, 5; 5, 3]
Square a 2 x 2
[2 1; 0 3]^2 = [4 5; 0 9]
A^k
[4, 5; 0, 9]
Zero exponent
A^0 = I even for a singular matrix
A^k
[1, 0; 0, 1]
Negative exponent
[1 1; 1 0]^-3 = [-1 2; 2 -3]
A^k
[-1, 2; 2, -3]
Large exponent by squaring
[1 1; 1 0]^20 contains F(21) = 10946
A^k
[10946, 6765; 6765, 4181]
Negative power of a singular matrix
[1 2; 2 4]^-1 does not exist
A^k
Error
Frequently asked questions
How do you square a matrix?+
Multiply it by itself using matrix multiplication: A^2 = A·A. Do not square the individual entries.
What is a matrix to the power of 0?+
The identity matrix I of the same size, for every square matrix, including singular ones.
Can you raise a matrix to a negative power?+
Only if it is invertible. A^(-k) = (A⁻¹)^k. If det(A) = 0 there is no inverse and no negative power.
Can I raise a non-square matrix to a power?+
No. A·A requires the number of columns of A to equal its number of rows, so only square matrices have powers.
Why does the Fibonacci matrix work?+
Multiplying [F(n+1) F(n); F(n) F(n-1)] by [1 1; 1 0] applies the recurrence F(n+2) = F(n+1) + F(n), so each power advances the sequence one step.
Is (A·B)^k equal to A^k·B^k?+
Only when A and B commute. In general (AB)^2 = ABAB, which is not AABB.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026