Description
Compute trace and determinant, then return the four row-major entries of A² - trace(A)A + det(A)I.
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 and determinant, then return the four row-major entries of A² - trace(A)A + det(A)I.
When to use this tool
Use Matrix 2x2 Cayley Hamilton Residual 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 residual. Use the output convention below.
Compute trace and determinant, then return the four row-major entries of A² - trace(A)A + det(A)I.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 |
|---|---|---|
| residual | Signed residual using the convention in the calculation paragraph. | Required. Array of finite number; maximum items 256. |
{
"input": {
"a": 1,
"b": 2,
"c": 3,
"d": 4
},
"output": {
"residual": [
0,
0,
0,
0
]
}
}Limitations and assumptions
- Exact algebra gives zero but floating-point residuals may remain; no tolerance-based theorem check or conditioning analysis is applied.
- 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.
- Principal Two By Two Minor
Compute a × d - b × c for the supplied two-by-two minor.