Compute the classical adjoint (adjugate) matrix with cofactors, minors, sign pattern visualization, inverse connection, and verification that A·adj(A) = det(A)·I.
The adjugate (classical adjoint) of a square matrix is the transpose of its cofactor matrix. It provides an elegant formula for the matrix inverse: A⁻¹ = adj(A)/det(A), and satisfies the fundamental identity A·adj(A) = det(A)·I regardless of whether A is invertible.
To compute the adjugate, three steps are required. First, compute the matrix of minors: Mᵢⱼ is the determinant of the submatrix obtained by deleting row i and column j. Second, apply the checkerboard sign pattern to get cofactors: Cᵢⱼ = (−1)^(i+j)·Mᵢⱼ. Third, transpose the cofactor matrix to obtain the adjugate: adj(A) = Cᵀ.
The adjugate is deeply connected to the determinant and inverse. Even when a matrix is singular (det = 0), the adjugate still exists — it just cannot be used to compute the inverse. For invertible matrices, the adjugate provides Cramer's rule and the cofactor expansion formula for determinants.
This calculator handles matrices from 2×2 up to 5×5, displaying the full minors matrix, cofactor matrix with sign pattern, the adjugate, and verification that A·adj(A) = det(A)·I. For invertible matrices, it also computes A⁻¹ = adj(A)/det(A).
Computing the adjugate requires calculating n² minor determinants, applying the checkerboard sign pattern, and transposing the result — a process that produces cascading arithmetic errors even for 3×3 matrices. This calculator displays the full minors matrix, cofactor matrix, and adjugate side by side, verifying that A·adj(A) = det(A)·I. It is invaluable for students learning cofactor expansion, anyone deriving matrix inverses via the adjugate formula, and engineers who need quick verification of symbolic matrix computations.
adj(A) = Cᵀ, where C is the cofactor matrix with Cᵢⱼ = (−1)^(i+j)·det(Mᵢⱼ). A⁻¹ = adj(A)/det(A).
Result: det(A) = 22, adj(A) = [[24,−12,−2],[5,3,−5],[−4,2,4]]
Computing cofactors and transposing gives the adjugate. A·adj(A) = 22·I confirms the result.
Computing the adjugate of an n×n matrix follows a precise three-step recipe. First, build the **matrix of minors**: for each position (i,j), delete row i and column j, then compute the determinant of the remaining (n−1)×(n−1) submatrix. Second, apply the **cofactor sign pattern**: multiply each minor by (−1)^(i+j), producing the cofactor matrix C. Third, **transpose** C to obtain adj(A) = Cᵀ. For a 2×2 matrix [[a,b],[c,d]], the adjugate simplifies to [[d,−b],[−c,a]] — swap the diagonals and negate the off-diagonals.
The adjugate provides the classical formula A⁻¹ = adj(A)/det(A), valid whenever det(A) ≠ 0. This formula underpins Cramer's Rule, where each solution variable xᵢ is expressed as a ratio of determinants. Even for singular matrices (det = 0), the identity A·adj(A) = det(A)·I still holds — it just produces the zero matrix on the right side. The adjugate also satisfies det(adj(A)) = det(A)^(n−1), a useful identity for verifying computations.
While LU decomposition is preferred for numerical matrix inversion, the adjugate formula remains essential in symbolic algebra (deriving closed-form inverses), control theory (transfer function matrices), and differential equations (matrix exponentials). The cofactor matrix itself appears in the derivative of the determinant with respect to matrix entries: ∂det(A)/∂aᵢⱼ = Cᵢⱼ, connecting the adjugate to sensitivity analysis and optimization. Understanding the adjugate deepens insight into how determinants, inverses, and linear systems are all interconnected.
In linear algebra, "adjoint" can mean two things: the classical adjoint (adjugate), which is the transpose of the cofactor matrix, or the conjugate transpose (Hermitian adjoint) A*. This calculator computes the classical adjoint/adjugate.
This follows from cofactor expansion. The diagonal entries of A·adj(A) are cofactor expansions along each row (giving det(A)), while off-diagonal entries are cofactor expansions with mismatched rows (giving 0).
The adjugate exists for singular matrices, but you cannot divide by det(A) = 0 to get the inverse. The identity A·adj(A) = 0·I = O still holds.
Cramer's rule states that xᵢ = det(Aᵢ)/det(A), which is equivalent to x = A⁻¹b = adj(A)b/det(A). Each component of adj(A)b gives the numerator determinant.
The cofactor Cᵢⱼ = (−1)^(i+j)·Mᵢⱼ, creating a checkerboard of + and − signs: the top-left is always +, and it alternates from there. Use this as a practical reminder before finalizing the result.
No — it requires computing n² determinants of (n−1)×(n−1) submatrices, giving O(n²·n!) complexity via cofactor expansion. For practical computation of inverses, LU decomposition is preferred.