Random Determinant Sign Transformation Quiz Generator

Description

Draw a supported row swap or scaling and retain its determinant sign rule, inheriting the delegated determinant core's 12-significant-digit rounding.

A matrix construction produces numeric arrays with a target algebraic property. Floating-point residuals check how closely a computed result satisfies that property, while conditioning determines how sensitive it is to numerical error.

Draw a supported row swap or scaling and retain its determinant sign rule, inheriting the delegated determinant core's 12-significant-digit rounding.

When to use this tool

Use this generator to create reproducible determinant sign transformation quiz cases from the supplied model or bank. Inspect the returned evidence before using cases in a test or exercise.

How to use the tool and how it works

Supply matrix, operation in the field shapes below. Set seed when you need an explicit replay.
Run the tool and correct any input error. For generated controls, distinguish a successful fixture containing an invalid case from failure to generate the fixture.
Read before, after, matrix, operation, multiplier, signChanged. Use the conventions below. Keep answer-bearing fields separate when presenting an exercise.

Apply one sampled elementary row transformation of the requested kind, calculate determinants before and after, and report its determinant multiplier and whether two nonzero determinants change sign.^[1]

Input field	Meaning and units	Accepted input
matrix	Matrix	Required. Array of Array of Safe integer.; minimum items 2; maximum items 3.; minimum items 2; maximum items 3.
operation	Operation	Required. Text; one of "swap", "scale".
seed	Optional numeric seed; takes precedence over context randomness. Without either, use the fixed tool.io default seed.	Optional. Safe integer.

Output field	Interpretation	Returned shape
before	Before	Required. Finite number.
after	After	Required. Finite number.
matrix	Matrix	Required. Array of Array of Safe integer.; minimum items 2; maximum items 3.; minimum items 2; maximum items 3.
operation	Operation	Required. Object; no unknown keys.
operation.kind	Kind	Required. Text; one of "swap", "scale", "add".
operation.row	Row	Required. Safe integer.
operation.other	Other	Required. Safe integer.
operation.factor	Factor	Required. Safe integer.
multiplier	Multiplier	Required. Finite number.
signChanged	Sign Changed	Required. Boolean.
randomness	Replay source and normalized 32-bit seed, or null seed for an advancing context source.	Required. Object; no unknown keys.
randomness.seed	Optional safe-integer replay seed; normalization and precedence are explained below.	Required. Safe integer. or Null.
randomness.source	Source	Required. Text; one of "context", "seed".

An explicit safe-integer seed takes precedence over injected context randomness and is normalized to an unsigned 32-bit integer. With no seed and no context source, replay uses the fixed tool.io default seed. A context source must return finite values from zero inclusive to one exclusive; its advancing state must be retained separately. randomness records the source and normalized seed, or null for context. These generators are not cryptographic random sources.

json

{
  "input": {
    "matrix": [
      [
        2,
        0
      ],
      [
        0,
        3
      ]
    ],
    "operation": "swap",
    "seed": 42
  },
  "output": {
    "before": 6,
    "after": -6,
    "operation": {
      "kind": "swap",
      "row": 0,
      "other": 1,
      "factor": -1
    },
    "matrix": [
      [
        0,
        3
      ],
      [
        2,
        0
      ]
    ],
    "multiplier": -1,
    "signChanged": true,
    "randomness": {
      "seed": 42,
      "source": "seed"
    }
  }
}

Limitations and assumptions

Zero determinants have signChanged false. Scaling by a negative factor can flip sign while row addition does not; rows and intermediates obey safe-integer bounds and determinant calculations inherit numeric precision.
A replay requires the same complete input and implementation. The stated sampling convention applies within its eligible space. A finite draw does not exhaust all possible outputs; repeats remain possible unless a distinct-selection rule is explicit.
The field tables state structural bounds; cross-field constraints and bounded search or output budgets also apply. Unknown keys are rejected where the object contract disallows them. Input errors are distinct from null fields and deliberate invalid controls in a successful output.

Alternative or Complementary methods

For a supplied example, inspect the returned construction or delegated check directly. To evaluate a consumer, also test cases chosen for its own boundaries; a generated finite corpus does not establish exhaustive coverage.

References

Tool behavior and example: Random Determinant Sign Transformation Quiz Generator — tool.io

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