The formula this page uses
p(λ) = det(A − λI) for 2×2: p(λ) = λ² − (trace A)·λ + det A
← Back to calculators
Characteristic Polynomial Calculator
Solve characteristic polynomial problems with clear steps, notation, and a final check.
x²
Calculate without using AI.
Evaluate the governing formula locally in your browser. Define each known quantity once, then change values to test another case instantly.
Choose an example or enter a formula.
Show your work Step-by-step method
- No account, AI request, or paid token is used.
Recent calculations
No calculations yet.
Algebra
Characteristic Polynomial Calculator explained
The short version
- The characteristic polynomial is what you get when you compute det(A − λI) and leave λ as a letter.
- Setting it to zero gives the eigenvalues: the stretch factors along the directions a matrix does not turn.
- For any 2×2 matrix there is a shortcut: λ² − (trace)λ + (determinant).
What each part means
SymbolWhat it means
A — The square matrixMust have the same number of rows as columns, or det(A − λI) is not defined.
λ — EigenvalueThe unknown stretch factor. A pure number with no unit.
I — Identity matrix1s down the diagonal, 0s elsewhere, so λI subtracts λ from each diagonal entry only.
trace A — TraceThe sum of the diagonal entries. It equals the sum of the eigenvalues.
Show your work: a full example
- The matrixA = [[4, 1], [2, 3]]
- Subtract λ down the diagonalA − λI = [[4 − λ, 1], [2, 3 − λ]]
- Take the determinant(4 − λ)(3 − λ) − (1)(2)
- Expand12 − 7λ + λ² − 2 = λ² − 7λ + 10
- Cross-check with the shortcuttrace = 4 + 3 = 7 and det = 12 − 2 = 10, giving λ² − 7λ + 10
- Set it to zero and factor(λ − 2)(λ − 5) = 0, so λ = 2 and λ = 5
- Check2 + 5 = 7 matches the trace and 2 × 5 = 10 matches the determinant
A second, different case
- A different case: a quarter-turn matrixB = [[0, −1], [1, 0]]
- Trace0 + 0 = 0
- Determinant(0)(0) − (−1)(1) = 1
- Apply the shortcutp(λ) = λ² − 0·λ + 1 = λ² + 1
- Solveλ² = −1, so λ = i and λ = −i
- Check the discriminant0² − 4(1)(1) = −4, negative, so no real eigenvalue exists
- Read it geometricallyB rotates every vector by 90°, so no direction is left pointing where it started
Copy-ready example
A = [[4, 1], [2, 3]]
The matrix
Questions about the Characteristic Polynomial Calculator
Why is it called the characteristic polynomial?
Because it is a fingerprint that survives a change of basis. Two matrices that describe the same transformation in different coordinate systems produce the identical polynomial, so its coefficients describe the transformation rather than the numbers you happened to write down.
Does row reducing A first make the calculation easier?
No, it breaks it. Row operations change eigenvalues, so the polynomial you get afterwards belongs to a different matrix. Expand det(A − λI) directly, or use the trace-and-determinant shortcut for a 2×2.
Is the 2×2 shortcut worth memorising?
Yes, it turns a determinant expansion into two glances. Sum the diagonal for the trace, cross-multiply for the determinant, and write λ² − (trace)λ + det. The 3×3 version exists too but involves the sum of the three 2×2 minors.
Can two different matrices have the same characteristic polynomial?
Yes. [[2, 0], [0, 2]] and [[1, 1], [0, 1]] are genuinely different transformations, yet the table above shows both give a polynomial with a single repeated root. The polynomial pins down eigenvalues but not the full behaviour.