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Characteristic Polynomial Calculator.

Find the characteristic polynomial coefficients for a 2×2 matrix.

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Trace: 5

Trace

5.000000
Determinant: 5

Determinant

5.000000
λ coefficient: -5

λ coefficient

-5.000000
Constant: 5

Constant

5.000000

Results update automatically as you type.

Use Cases

Verify matrix properties

The coefficients give the trace (sum of diagonal entries) and determinant. This helps quickly check if a matrix is singular (det=0) or to compute invariants under similarity transformations.

Example: For matrix [[1,2],[3,4]], trace=5, det=-2, so polynomial is λ² - 5λ - 2.

Frequently Asked Questions

What is the characteristic polynomial of a 2x2 matrix?
For a 2x2 matrix A = [[a11, a12], [a21, a22]], the characteristic polynomial is p(λ) = det(λI - A) = λ² - (a11 + a22)λ + (a11*a22 - a12*a21). The coefficients are: 1 (for λ²), -(trace) for λ, and determinant as the constant term.
How do I use this calculator?
Enter the four entries of your 2x2 matrix: a11 (top-left), a12 (top-right), a21 (bottom-left), a22 (bottom-right). The calculator will output the coefficients of the characteristic polynomial in the form λ² + bλ + c, where b = -(a11+a22) and c = a11*a22 - a12*a21.
What are the eigenvalues and how are they related?
The eigenvalues of the matrix are the roots of the characteristic polynomial. For a 2x2 matrix, they are given by λ = (tr(A) ± sqrt(tr(A)² - 4*det(A)))/2. The polynomial coefficients directly give you the trace and determinant, which are key to finding eigenvalues.

Tips & Common Mistakes

Tips

  • Ensure you enter numbers only; the calculator expects numeric values for each matrix entry.
  • Double-check the order: a11 is top-left, a12 top-right, a21 bottom-left, a22 bottom-right.
  • The characteristic polynomial is always monic (leading coefficient 1) for a 2x2 matrix.
  • Use the result to quickly find eigenvalues by solving the quadratic equation.

Common Mistakes to Avoid

  • Mixing up the signs: the coefficient of λ is negative of the trace (a11+a22), not the trace itself.
  • Forgetting to subtract the product of off-diagonal entries when computing the determinant (a11*a22 - a12*a21).
  • Entering values in the wrong positions, which changes the trace and determinant and thus the polynomial.

Last updated: August 13, 2026