Adjugate matrix
In linear algebra, the adjugate matrix (also known as the classical adjoint) of an n \times n square matrix A is defined as the transpose of its cofactor matrix C(A), where the (i,j)-th entry of C(A) is the cofactor C_{ij} = (-1)^{i+j} \det(M_{ij}) and M_{ij} is the (n-1) \times (n-1) submatrix obtained by deleting the i-th row and j-th column of A.[1][2] The resulting adjugate, denoted \operatorname{adj}(A), has entries that are signed determinants of these minors, providing a matrix whose structure encodes information about the original matrix's subdeterminants without requiring division operations.[3] A fundamental property of the adjugate is the relation A \cdot \operatorname{adj}(A) = \operatorname{adj}(A) \cdot A = \det(A) \, I_n, where I_n is the n \times n identity matrix.[1] For a nonsingular matrix (i.e., \det(A) \neq 0), this implies that the inverse matrix is given explicitly by A^{-1} = \frac{1}{\det(A)} \operatorname{adj}(A), making the adjugate a key tool for computing inverses via cofactor expansion, though this method is computationally inefficient for large n > 2 compared to alternatives like Gaussian elimination.[2][4] The adjugate exists even for singular matrices, where it satisfies \operatorname{adj}(A) = 0 if \operatorname{rank}(A) < n-1. The terminology "adjugate" was adopted to distinguish this construction from the "adjoint" in other contexts, such as the conjugate transpose (Hermitian adjoint) for complex matrices or the adjoint operator in functional analysis.[3] The adjugate facilitates applications beyond inversion, including Cramer's rule for solving linear systems A \mathbf{x} = \mathbf{b} (where x_i = \det(A_i)/\det(A) and A_i replaces the i-th column of A with \mathbf{b}) and generalizations in multilinear algebra.[4]Fundamentals
Definition
The adjugate matrix of an n \times n matrix A over a commutative ring with identity, denoted \operatorname{adj}(A), is defined as the transpose of the cofactor matrix C(A), where the entry C(A)_{i,j} = (-1)^{i+j} \det(M_{i,j}) and M_{i,j} is the submatrix of A obtained by deleting the i-th row and j-th column (the (i,j)-minor).[1] This construction arises naturally from the cofactor expansion formula for the determinant, which states that for any fixed row k, \det(A) = \sum_{m=1}^n a_{k m} C(A)_{k m}, where C(A)_{k m} = (-1)^{k+m} \det(M_{k,m}) is the cofactor along that row; the adjugate collects these cofactors into a matrix whose transpose encodes the expansion coefficients across all rows and columns, facilitating matrix-level identities.[2] A fundamental consequence of this definition is the identity A \cdot \operatorname{adj}(A) = \operatorname{adj}(A) \cdot A = \det(A) I_n, where I_n is the n \times n identity matrix; to see this, consider the (i,j)-entry of A \cdot \operatorname{adj}(A), which equals \sum_{k=1}^n a_{i k} C(A)_{j k}. If i = j, this is the cofactor expansion of \det(A) along row i; if i \neq j, it equals the cofactor expansion along row i of the matrix obtained by replacing row j with row i, which has two identical rows and thus determinant zero.[2]Notation and Cofactors
The adjugate of a square matrix A, often denoted \operatorname{adj}(A) or \operatorname{adjugate}(A), is the transpose of the cofactor matrix of A; alternative notations include A^{\mathrm{co}} for the cofactor matrix itself or the classical term "adjugata" in historical contexts.[5][6] This notation must be distinguished from the "adjoint" in modern linear algebra, which typically refers to the conjugate transpose A^* or A^H for complex matrices, whereas the adjugate pertains specifically to cofactor-based construction.[7][8] The cofactor matrix of A = (a_{ij}) is constructed entrywise from the cofactors C_{ij}, where each C_{ij} = (-1)^{i+j} \det(M_{ij}) and M_{ij} is the (n-1) \times (n-1) minor submatrix obtained by deleting the i-th row and j-th column of A.[5][9] The sign alternation (-1)^{i+j} ensures the cofactor expansion aligns with the determinant formula along any row or column.[6] Minors represent the determinants of these submatrices, providing the foundational building blocks for the entire structure over fields or rings where determinants are defined.[5] Cofactors are typically computed via Laplace expansion, which recursively applies the determinant formula to each minor: for the determinant along the first row, \det(A) = \sum_{j=1}^n a_{1j} C_{1j}, and this process repeats for submatrices.[9] This method is efficient for small dimensions (n \leq 3) but exhibits factorial time complexity O(n!) for general n \times n matrices, as each of the n^2 cofactors requires computing an (n-1) \times (n-1) determinant, leading to exponential growth in operations.[10][11] For any square matrix over a commutative ring, the adjugate is uniquely determined, as the construction relies solely on the well-defined minors and their signed determinants, which exist uniquely in such algebraic structures.[12][13]Examples
Low-Dimensional Cases
For the simplest case of a $1 \times 1 matrix A = , the adjugate is \operatorname{adj}(A) = {{grok:render&&&type=render_inline_citation&&&citation_id=1&&&citation_type=wikipedia}}, as the determinant \det(A) = a and the relation A \cdot \operatorname{adj}(A) = \det(A) I_1 holds trivially with the $1 \times 1 identity matrix I_1 = {{grok:render&&&type=render_inline_citation&&&citation_id=1&&&citation_type=wikipedia}}.[14] Consider a general $2 \times 2 matrix A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}. The cofactor matrix C(A) has entries C_{11} = d, C_{12} = -c, C_{21} = -b, and C_{22} = a, so C(A) = \begin{pmatrix} d & -c \\ -b & a \end{pmatrix}. The adjugate is the transpose of this cofactor matrix: \operatorname{adj}(A) = C(A)^T = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}.[14] To verify the defining property, compute the product A \cdot \operatorname{adj}(A) = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} = \begin{pmatrix} ad - bc & 0 \\ 0 & ad - bc \end{pmatrix} = (ad - bc) I_2, where I_2 is the $2 \times 2 identity matrix and \det(A) = ad - bc.[14] If \det(A) \neq 0, the inverse matrix is given by A^{-1} = \frac{1}{\det(A)} \operatorname{adj}(A) = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}, with explicit entries \frac{d}{ad - bc}, \frac{-b}{ad - bc}, \frac{-c}{ad - bc}, and \frac{a}{ad - bc}.[14] In this low-dimensional case, the entries of the adjugate are precisely the complementary elements of A along the main anti-diagonal, with appropriate sign changes for the off-diagonal positions.[14]Higher-Dimensional Illustration
To illustrate the adjugate for a matrix beyond 2×2 dimensions, consider a general 3×3 matrix A = (a_{ij})_{i,j=1}^3. The cofactor C_{ij} is defined as (-1)^{i+j} times the determinant of the 2×2 submatrix obtained by removing the i-th row and j-th column of A. The adjugate \operatorname{adj}(A) is then the transpose of the 3×3 matrix formed by these cofactors, so each entry of \operatorname{adj}(A) is itself a signed 2×2 determinant from a minor of A.[15] For a concrete numerical example, take the symmetric tridiagonal matrix A = \begin{pmatrix} 2 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 2 \end{pmatrix}. The cofactor matrix C = (C_{ij}) is computed as follows:- C_{11} = (+1) \det \begin{pmatrix} 2 & -1 \\ -1 & 2 \end{pmatrix} = 3,
- C_{12} = (-1) \det \begin{pmatrix} -1 & -1 \\ 0 & 2 \end{pmatrix} = 2,
- C_{13} = (+1) \det \begin{pmatrix} -1 & 2 \\ 0 & -1 \end{pmatrix} = 1,
- C_{21} = (-1) \det \begin{pmatrix} -1 & 0 \\ -1 & 2 \end{pmatrix} = 2,
- C_{22} = (+1) \det \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix} = 4,
- C_{23} = (-1) \det \begin{pmatrix} 2 & -1 \\ 0 & -1 \end{pmatrix} = 2,
- C_{31} = (+1) \det \begin{pmatrix} -1 & 0 \\ 2 & -1 \end{pmatrix} = 1,
- C_{32} = (-1) \det \begin{pmatrix} 2 & 0 \\ -1 & -1 \end{pmatrix} = 2,
- C_{33} = (+1) \det \begin{pmatrix} 2 & -1 \\ -1 & 2 \end{pmatrix} = 3.