{"id":"determinant","domain":"algebra","type":"Concept","title":"Determinant","slug":"determinant","url":"/mathematics/algebra/determinant/","summary":"A single number from a square matrix revealing invertibility and area/volume scaling factor. Zero determinant means the transformation collapses dimensions.","formal_definition":"Leibniz formula: sum over all permutations sigma of sgn(sigma) times product of a_i,sigma(i)","added_to_codex":"2026-05-27","last_verified":"2026-05-27","sources":[{"tier":1,"citation":"Strang, G. (2016). Introduction to Linear Algebra. 5th ed."},{"tier":1,"citation":"Axler, S. (2015). Linear Algebra Done Right. 3rd ed. Springer."},{"tier":2,"citation":"Kline, M. (1972). Mathematical Thought from Ancient to Modern Times. Oxford University Press."},{"tier":3,"citation":"Muir, T. (1906-1930). The Theory of Determinants in the Historical Order of Development. 4 vols."}],"relationships":[{"type":"computed_from","target":"matrix","label":"Matrix"},{"type":"uses","target":"integral","label":"Integral — Jacobian"},{"type":"related_to","target":"vector-space","label":"Vector Space"}],"entry_status":"live","reading_levels_available":["curious","exploring","deep_dive"],"completeness":1.0,"codex_version":"1.7"}