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Characteristic polynomial calculator

Enter a square matrix (2×2 to 5×5) and get its characteristic polynomial p(λ) = det(λI − A), whose roots are the eigenvalues. It uses the Faddeev–LeVerrier algorithm, which is exact at any size, and also reports the trace and determinant.

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Characteristic polynomial calculator is free to use as often as you like, directly from this page. Its place is under Algebra & equations; Polynomial Factoring Calculator and Polynomial Long Division Calculator answer the questions closest to this one.

How to use it

  1. Open the tool — no signup or install needed.
  2. Enter your input or adjust the available options.
  3. Get your result instantly, then copy or download it.

Frequently asked questions

What does Characteristic polynomial calculator do?

Enter a square matrix (2×2 to 5×5) and get its characteristic polynomial p(λ) = det(λI − A), whose roots are the eigenvalues. It uses the Faddeev–LeVerrier algorithm, which is exact at any size, and also reports the trace and determinant.

What does a concrete case look like?

[[2, 1], [1, 2]] → λ² − 4λ + 3 → λ = 1, λ = 3 — the tool shows every step in between, not just the final figure.

When would I actually use this?

Getting through a problem set: solving for the unknown, factoring an expression, and — more usefully — seeing the steps that got there.

What is the most common mistake?

Losing a solution when both sides are squared or divided by an expression. Squaring can add roots that do not satisfy the original equation, and dividing can remove one — check every answer back in the original.

How is Characteristic polynomial calculator different from Polynomial Factoring Calculator?

They sit next to each other but answer different questions: Polynomial Factoring Calculator is the one to open when you need it to factor over ℚ with the pattern named at every step: GCF, difference of squares, perfect square, sum or difference of cubes (SOAP), the ac-method and grouping — then verified by re-expansion. Pick whichever matches what you're starting from — both are free.

Is there a tool for the next step?

Polynomial Long Division Calculator is the closest one after this: The whole tableau, subtraction row by subtraction row, with quotient, remainder and the identity dividend = divisor × quotient + remainder checked for you.

What else is worth having open alongside it?

Polynomial Roots Calculator and Euler characteristic calculator (V − E + F) — they come up in the same task often enough to be worth a second tab.

Where do the figures come from?

The algebra is symbolic where the expression allows it and numeric otherwise, and the tool says which it used. Exact results are given as fractions or radicals rather than rounded decimals.

Further reading

All guides →
How-toHow to Use the Quadratic Formula: Solving ax² + bx + c = 0Solve any quadratic equation with x = (−b ± √(b² − 4ac)) / 2a, and read the discriminant to know how many real solutions to expect.How-toHow to Find a Cube Root by HandA step-by-step guide to finding a cube root by estimation and refinement — plus how it differs from a square root and when a value stays irrational.ExplainerHow Exponents Work: Powers and the Laws of ExponentsAn exponent is repeated multiplication. Learn the laws of exponents, what negative and fractional powers mean, and why anything to the zero is one.ExplainerThe Flat Pattern of a Cone Is a Sector, Not a CircleRoll a 50 mm cone 80 mm tall and its pattern is a 190.8° slice of a 94 mm circle. Flatten a nearly cylindrical lampshade and the pattern radius runs to 839 mm — the shallower the taper, the further away the apex.ExplainerA 1 % Drop Rate Does Not Mean One Hundred RunsAt 1 %, a hundred attempts give you 63.4 % — not certainty. Ninety per cent takes 230 attempts and ninety-nine takes 459, and more than a third of players are still empty-handed at a hundred.ExplainerFibonacci and the Golden RatioThe Fibonacci sequence adds each pair of terms to make the next; the ratio of neighboring terms homes in on the golden ratio φ ≈ 1.618. See how, and where the pattern turns up.