How this calculation works
Repeated squaring computes an integer matrix power. Negative powers first require an inverse. A zero exponent returns the identity by the matrix-power convention. This operation raises the matrix as a linear operator, not each entry separately.
Aᵏ = repeated matrix multiplication; A⁰ = I; A⁻ᵏ = (A⁻¹)ᵏA worked example
The fifth power of [1 1; 1 0] is [8 5; 5 3]. Its entries follow adjacent Fibonacci numbers; this is a consequence of repeatedly applying that specific matrix.
Good to know
Exact rational matrices up to 6 × 6. Entries are bounded to ±1,000,000, integer fraction denominators to 1,000,000 and decimal input to 6 places. Intermediate fractions have a size limit. No complex entries, symbolic expressions, measurement uncertainty or numerical-conditioning analysis. Ordered vectors are entered as matrix columns.
Reference: Harvey Mudd College: matrix algebra (opens in a new tab) ↗
Repeated matrix multiplication represents repeated action
A² means A multiplied by A using row-by-column multiplication. It does not square each entry. If A describes one step of a linear process, Aᵏ describes k repeated steps using the same operator.
The matrix [1 1; 1 0] advances a pair of consecutive Fibonacci numbers. Its fifth power is [8 5; 5 3]. The pattern follows from the recurrence embedded in the matrix, and does not apply to arbitrary matrices.
Only square matrices can be repeatedly multiplied by themselves. A rectangular matrix might map between spaces of different dimensions, so its output cannot automatically be fed back into the same matrix.
Matrix multiplication defines each repeated step. Squaring a matrix is generally different from squaring its entries one by one.
Zero and negative powers have separate requirements
The matrix-power convention sets A⁰ to the matching identity, including when A is singular. The identity is the neutral element for multiplication. This convention should not be confused with an attempt to divide a matrix by itself.
A negative exponent requires A to be invertible. For example, A⁻³ means multiply A⁻¹ by itself three times. A singular matrix has no ordinary inverse and is rejected for negative exponents.
Repeated squaring combines powers efficiently: A⁸ can be obtained by squaring A, then A², then A⁴. Exact rational values are retained, although a result whose fractions grow beyond the supported size limit will stop with a clear error. Fractional powers and the matrix exponential are separate operations.
A few questions, answered
Can I enter a fractional exponent?
No. This page accepts integer exponents from −20 to 20. Fractional matrix powers require additional conventions and are not computed here.
Will A² always have larger entries?
No. A rotation, projection, nilpotent matrix or fractional scaling can behave very differently. Matrix powers do not imply entrywise growth.