{"id":"matrix-solve-linear","version":"1.0.0","description":"Solve the small square linear system A x = b for x using Gaussian elimination.","supported_operations":["solve linear system","solve Ax=b","linear equation solver"],"unsupported_operations":["non-square or rank-deficient (singular) systems","systems larger than 12x12","least-squares solutions for overdetermined systems"],"semantics":["Input a is a square NxN coefficient matrix of finite JSON numbers, N from 1 to 12. Input b is a length-N finite JSON number vector. Magnitudes are bounded to at most 1000000.","Computed via Gaussian elimination with partial (largest-magnitude) pivoting and back substitution, in IEEE-754 double precision.","Optional input tolerance (default 1e-9, range 0 to 1) is the minimum acceptable pivot magnitude; if every remaining candidate pivot in a column is at or below tolerance, the system is rejected as singular or numerically singular (no unique solution).","The result is the unique solution vector x such that a * x approximately equals b; a result entry of -0 is normalized to 0.","Every result entry is checked to be a finite number before it is returned; a very small nonzero pivot (below tolerance-rejection but large enough to be accepted) can otherwise amplify through elimination to overflow, which is rejected as invalid input rather than silently serialized as null."],"limits":{"max_dim":12,"max_abs_value":1000000},"pricing":{"status":"unpriced","charge_usd":null},"input_schema":{"type":"object","additionalProperties":false,"required":["a","b"],"properties":{"a":{"type":"array","minItems":1,"maxItems":12,"items":{"type":"array","items":{"type":"number"}}},"b":{"type":"array","minItems":1,"maxItems":12,"items":{"type":"number"}},"tolerance":{"type":"number","minimum":0,"maximum":1}}},"output_schema":{"type":"object","additionalProperties":false,"required":["result"],"properties":{"result":{"type":"array","items":{"type":"number"}}}},"examples":[{"input":{"a":[[2,1],[1,3]],"b":[3,5]},"output":{"result":[0.8,1.4]}}],"execute_url":"/v1/tools/matrix-solve-linear/versions/1.0.0/execute"}