Description
Return trace a + d as the eigenvalue sum and determinant ad - bc as their product.
A determinant encodes signed volume scaling and changes predictably under elementary row or column operations. Small-matrix identities depend on entry order and exact algebraic assumptions.
Return trace a + d as the eigenvalue sum and determinant ad - bc as their product.
When to use this tool
Use Matrix 2x2 Eigenvalue Sum Product when you need one determinant identity, sign, or small-matrix diagnostic.
How to use the tool and how it works
- Enter a, b, c, d. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read sum, product. Use the output convention below.
Return trace a + d as the eigenvalue sum and determinant ad - bc as their product.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| a | A | Required. Finite number. |
| b | B | Required. Finite number. |
| c | C | Required. Finite number. |
| d | D | Required. Finite number. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| sum | Sum | Required. Finite number. |
| product | Product | Required. Finite number. |
json
{
"input": {
"a": 1,
"b": 2,
"c": 3,
"d": 4
},
"output": {
"sum": 5,
"product": -2
}
}Limitations and assumptions
- The identities include complex eigenvalue pairs; individual eigenvalues, ordering, and multiplicity are not calculated.
- Inputs must satisfy both the field types and the cross-field conditions above. Invalid input causes an error; a successful false flag or documented placeholder is a different result.
- Numeric calculations use binary floating point unless exact decimal-integer arithmetic is explicitly stated. No hidden measurement tolerance or decimal rounding is added; nonfinite or out-of-contract results are rejected. Numeric signed-zero results are exported as zero.
Alternative or Complementary methods
Use a stable factorization for full determinants and a scale-aware numerical analysis for near-singular matrices.
References
Similar or alternative tools
- Matrix Repeated Eigenvalue 2x2 Gap
Calculate discriminant = (a - d)^2 + 4bc.
- Matrix 2x2 Nilpotency Trace Determinant
Compute trace a + d and determinant ad - bc.