{"success":true,"version":"v1","request":{"tool":"get_calculator_schema","calculator_id":"matrix-inverse"},"result":{"entity_type":"calculator","id":"matrix-inverse","calculator_id":"matrix-inverse","canonical_url":"https://tttkmbb.com/math/matrix-inverse","name":"Matrix Inverse Calculator","title":"Matrix Inverse Calculator – 2×2 and 3×3 Inverse by the Adjugate Method (Cofactors and Determinant)","category":"math","category_name":"Math","tool_name":"calculate_matrix_inverse","featured_mcp_tool":false,"description":"Computes the inverse of a 2×2 or 3×3 matrix as adj(A) / det(A), returning the determinant, the cofactor and adjugate matrices and the inverse, with an explicit error for singular matrices.","use_when":"You need A⁻¹ for a small matrix, e.g. to solve A·x = b, undo a linear transformation, or check that A·A⁻¹ = I.","do_not_use_when":"You only need the determinant or rank (use matrix-determinant), want to solve a specific system (use linear-system), or the matrix is larger than 3×3.","inputs":[{"name":"matrix","label":"Matrix (row-major)","type":"number_list","required":true,"description":"All entries row by row: 4 values for 2×2 or 9 for 3×3. [[1, 2], [3, 4]] is entered as 1, 2, 3, 4.","example":[1,2,3,4]},{"name":"size","label":"Size n","type":"integer","required":false,"min":2,"max":3,"description":"Matrix dimension n (2 or 3). Optional: inferred from the number of entries when omitted.","example":2}],"outputs":[{"name":"determinant","label":"Determinant","type":"number","decimals":6,"description":"det(A); the inverse exists only when it is non-zero."},{"name":"inverse","label":"Inverse (row-major)","type":"number_list","decimals":6,"description":"Entries of A⁻¹ row by row."},{"name":"inverse_text","label":"Inverse","type":"string","decimals":4,"description":"A⁻¹ written as nested rows, e.g. '[[-2, 1], [1.5, -0.5]]'."},{"name":"adjugate","label":"Adjugate (row-major)","type":"number_list","decimals":6,"description":"adj(A) = transpose of the cofactor matrix; A⁻¹ = adj(A) / det(A)."},{"name":"cofactors","label":"Cofactor matrix (row-major)","type":"number_list","decimals":6,"description":"C_ij = (−1)^(i+j) · M_ij, where M_ij is the minor of entry (i, j)."},{"name":"size","label":"Size","type":"integer","decimals":0,"description":"Dimension n of the matrix."}],"input_schema":{"type":"object","properties":{"matrix":{"description":"All entries row by row: 4 values for 2×2 or 9 for 3×3. [[1, 2], [3, 4]] is entered as 1, 2, 3, 4.","type":"array","items":{"type":"number"},"minItems":4,"examples":[[1,2,3,4]]},"size":{"description":"Matrix dimension n (2 or 3). Optional: inferred from the number of entries when omitted.","type":"integer","minimum":2,"maximum":3,"examples":[2]}},"additionalProperties":false,"required":["matrix"]},"output_schema":{"type":"object","properties":{"determinant":{"description":"det(A); the inverse exists only when it is non-zero.","type":"number"},"inverse":{"description":"Entries of A⁻¹ row by row.","type":"array","items":{"type":"number"}},"inverse_text":{"description":"A⁻¹ written as nested rows, e.g. '[[-2, 1], [1.5, -0.5]]'.","type":"string"},"adjugate":{"description":"adj(A) = transpose of the cofactor matrix; A⁻¹ = adj(A) / det(A).","type":"array","items":{"type":"number"}},"cofactors":{"description":"C_ij = (−1)^(i+j) · M_ij, where M_ij is the minor of entry (i, j).","type":"array","items":{"type":"number"}},"size":{"description":"Dimension n of the matrix.","type":"integer"}}},"formula":"A⁻¹ = adj(A) / det(A), adj(A) = Cᵀ with cofactors C_ij = (−1)^(i+j)·M_ij; for 2×2: [[a, b], [c, d]]⁻¹ = 1/(ad − bc) · [[d, −b], [−c, a]]","sources":[{"name":"Wikipedia – Invertible matrix","url":"https://en.wikipedia.org/wiki/Invertible_matrix","type":"reference","retrieved_at":"2026-09-24"},{"name":"Wikipedia – Adjugate matrix","url":"https://en.wikipedia.org/wiki/Adjugate_matrix","type":"reference","retrieved_at":"2026-09-24"},{"name":"Wolfram MathWorld – Matrix Inverse","url":"https://mathworld.wolfram.com/MatrixInverse.html","type":"reference","retrieved_at":"2026-09-24"}],"freshness":{"type":"static","max_age_seconds":null,"note":"Deterministic formula with fixed constants; results never go stale. Inputs supplied by the caller determine the output."},"examples":[{"name":"[[1, 2], [3, 4]]","inputs":{"matrix":[1,2,3,4],"size":2},"expected":{"determinant":-2,"inverse":[-2,1,1.5,-0.5],"inverse_text":"[[-2, 1], [1.5, -0.5]]","adjugate":[4,-2,-3,1],"cofactors":[4,-3,-2,1],"size":2},"url":"https://tttkmbb.com/api/v1/calculate/matrix-inverse?matrix=1%2C2%2C3%2C4&size=2"},{"name":"[[1, 2, 3], [0, 1, 4], [5, 6, 0]]","inputs":{"matrix":[1,2,3,0,1,4,5,6,0]},"expected":{"determinant":1,"inverse":[-24,18,5,20,-15,-4,-5,4,1],"adjugate":[-24,18,5,20,-15,-4,-5,4,1],"size":3},"url":"https://tttkmbb.com/api/v1/calculate/matrix-inverse?matrix=1%2C2%2C3%2C0%2C1%2C4%2C5%2C6%2C0"}],"faq":[{"q":"Why does a singular matrix have no inverse?","a":"The inverse divides the adjugate by the determinant; when det(A) = 0 the matrix collapses space onto a line or plane and the mapping cannot be undone. The calculator then returns an error naming the zero determinant."},{"q":"How can I check the result?","a":"Multiply A by the inverse: the product must be the identity matrix (ones on the diagonal, zeros elsewhere), up to rounding of the 6 reported decimals."}],"tags":["matrix inverse","inverse matrix","3x3 inverse","adjugate","cofactor matrix","invert matrix"],"related":[{"calculator_id":"matrix-determinant","reason":"Determinant, rank and singularity test for the same matrix."},{"calculator_id":"linear-system","reason":"Solve A·x = b directly with Cramer's rule."}],"links":{"html":"https://tttkmbb.com/math/matrix-inverse","markdown":"https://tttkmbb.com/math/matrix-inverse.md","json":"https://tttkmbb.com/math/matrix-inverse.json","api":"https://tttkmbb.com/api/v1/calculate/matrix-inverse","schema":"https://tttkmbb.com/api/v1/calculators/matrix-inverse","openapi":"https://tttkmbb.com/openapi.json","mcp":"https://tttkmbb.com/mcp"},"version":"v1","updated_at":"2026-09-24"},"timestamp":"2026-09-24T01:47:03Z","next_actions":[{"tool":"run_calculator","calculator_id":"matrix-inverse","reason":"Run Matrix Inverse Calculator with the inputs above."}],"links":{"markdown":"https://tttkmbb.com/math/matrix-inverse.md"}}