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.
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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.