{"id":"matrix-determinant","version":"1.0.0","description":"Compute the determinant of a small square finite-number matrix.","supported_operations":["matrix determinant","det of a square matrix"],"unsupported_operations":["non-square matrices","matrices larger than 12x12","symbolic/exact-rational determinant"],"semantics":["Input matrix is a square NxN array of finite JSON numbers, N from 1 to 12, magnitude bounded to at most 1000000.","Computed via Gaussian elimination with partial (largest-magnitude) pivoting in IEEE-754 double precision, accumulating the product of pivots with a sign flip per row swap.","Results for small integer-valued matrices are typically exact; floating-point rounding can appear for larger, fractional, or ill-conditioned matrices.","If, at any elimination step, every remaining candidate pivot in a column is exactly 0, the determinant is exactly 0 (the input matrix is singular).","A result of -0 is normalized to 0."],"limits":{"max_dim":12,"max_abs_value":1000000},"pricing":{"status":"unpriced","charge_usd":null},"input_schema":{"type":"object","additionalProperties":false,"required":["matrix"],"properties":{"matrix":{"type":"array","minItems":1,"maxItems":12,"items":{"type":"array","items":{"type":"number"}}}}},"output_schema":{"type":"object","additionalProperties":false,"required":["result"],"properties":{"result":{"type":"number"}}},"examples":[{"input":{"matrix":[[1,2],[3,4]]},"output":{"result":-2}}],"execute_url":"/v1/tools/matrix-determinant/versions/1.0.0/execute"}