2x2 Characteristic Polynomial Calculator

Construct det(λI−A) in ascending coefficient order for real 2×2 input, not a generic trace/determinant alias; preserves full characteristic polynomial.

Description

The characteristic polynomial encodes a matrix's eigenvalues as roots.

The characteristic polynomial encodes a matrix's eigenvalues as roots. Constant-first coefficient ordering must be retained when passing the polynomial to another tool.^[1]

Construct det(λI−A) in ascending coefficient order for real 2×2 input, not a generic trace/determinant alias; preserves full characteristic polynomial.

When to use 2x2 Characteristic Polynomial Calculator

Use this tool to construct det(λI−A) in ascending coefficient order for real 2×2 input, not a generic trace/determinant alias; preserves full characteristic polynomial.

Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.

How to use 2x2 Characteristic Polynomial Calculator and understand its results

Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
Enter matrix using the field table and calculation rules, then run the tool.
Read characteristicPolynomialConstantFirst. The explanation below defines the units, signs, ordering and any flags or placeholder values.
If the tool reports an input issue, correct the values and run it again. Apply the stated assumptions when interpreting a successfully returned result.

Input field	Meaning	Value type; see rules below	Requirement
matrix	Matrix	list of list of number	Required

Enter a finite real 2×2 matrix. Return the ascending coefficient list [ad−bc,−a−d,1].

det(λI−A)=det(A)−trace(A)λ+λ²

Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.

Output field	Meaning	Value type
characteristicPolynomialConstantFirst	Characteristic polynomial constant first	list of number

json

{
  "input": {
    "matrix": [
      [
        1,
        2
      ],
      [
        3,
        4
      ]
    ]
  },
  "output": {
    "characteristicPolynomialConstantFirst": [
      -2,
      -5,
      1
    ]
  }
}

Limitations and assumptions

The polynomial does not itself return roots or eigenvectors. Extreme entries can cause nonfinite output and are rejected.

Numeric inputs and returned numeric values must be finite; an unrepresentable returned value produces an error. Matrices are limited to 128 rows and 128 entries per row; other numeric lists are limited to 100,000 entries. Missing required fields, extra fields, wrong value types and violations of the calculation rules are input errors.

Alternative or Complementary methods

Use 2x2 Eigenvalues Calculator: Compute both algebraic eigenvalues of a real 2×2 matrix, preserving complex conjugate pairs and exact computed repeated-root flag; no false real-only truncation or eigenvector independence claim. For a complementary task, use Discriminant Resultant Calculator: Compute exact degree-general integer polynomial discriminant from signed derivative resultant divided by leading coefficient; constant-first degree 1–12, no separate cubic/quartic aliases or real-root-count inference.

References

2x2 Characteristic Polynomial Calculator — tool.io

Similar or alternative tools

2x2 Eigenvalues Calculator
Compute both algebraic eigenvalues of a real 2×2 matrix, preserving complex conjugate pairs and exact computed repeated-root flag; no false real-only truncation or eigenvector independence claim.
Discriminant Resultant Calculator
Compute exact degree-general integer polynomial discriminant from signed derivative resultant divided by leading coefficient; constant-first degree 1–12, no separate cubic/quartic aliases or real-root-count inference.