Description
Compute trace a + d and determinant ad - bc.
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.
Compute trace a + d and determinant ad - bc.
When to use this tool
Use Matrix 2x2 Nilpotency Trace Determinant 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 nilpotent, trace, determinant. Inspect nilpotent before interpreting the associated values.
Compute trace a + d and determinant ad - bc. Mark nilpotent only when both are exactly zero.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 |
|---|---|---|
| nilpotent | Nilpotent | Required. Boolean. |
| trace | Trace | Required. Finite number. |
| determinant | Determinant | Required. Finite number. |
json
{
"input": {
"a": 0,
"b": 3,
"c": 0,
"d": 0
},
"output": {
"nilpotent": true,
"trace": 0,
"determinant": 0
}
}Limitations and assumptions
- This real two-by-two algebraic criterion has no numerical tolerance; it does not determine the nilpotency index or generalize to larger matrices.
- 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 2x2 Eigenvalue Sum Product
Return trace a + d as the eigenvalue sum and determinant ad - bc as their product.
- Matrix Repeated Eigenvalue 2x2 Gap
Calculate discriminant = (a - d)^2 + 4bc.