# Adjugate matrix

The adjugate matrix, also called the classical adjoint, of a square matrix \( A \) is the transpose of the matrix of cofactors of the elements of \( A \).<sup>[1](https://www.sciencedirect.com/science/article/pii/S0024379598100988)</sup> It satisfies the identity \( A\,(\operatorname{adj} A) = (\operatorname{adj} A)\,A = (\det A)\,I \), which makes it the algebraic building block of the inverse: when \( \det A \neq 0 \), \( A^{-1} = \operatorname{adj} A / \det A \).<sup>[2](https://scipp-legacy.pbsci.ucsc.edu/~haber/ph116A/adjugate.pdf)</sup>

| Key fact | Detail |
|---|---|
| Definition | \( (\operatorname{adj} A)_{ij} = C_{ji} \), the cofactor of the element at row \( j \), column \( i \); the adjugate is the transpose of the cofactor matrix<sup>[2](https://scipp-legacy.pbsci.ucsc.edu/~haber/ph116A/adjugate.pdf)</sup> |
| Defining identity | \( A\,(\operatorname{adj} A) = (\operatorname{adj} A)\,A = (\det A)\,I \)<sup>[2](https://scipp-legacy.pbsci.ucsc.edu/~haber/ph116A/adjugate.pdf)</sup> |
| Inverse formula | If \( \det A \neq 0 \), \( A^{-1} = \operatorname{adj} A / \det A \); if \( \det A = 0 \), the inverse does not exist<sup>[2](https://scipp-legacy.pbsci.ucsc.edu/~haber/ph116A/adjugate.pdf)</sup> |
| Determinant of the adjugate | \( \det(\operatorname{adj} A) = (\det A)^{n-1} \) for an \( n \times n \) matrix, a formula discovered by Cauchy<sup>[2](https://scipp-legacy.pbsci.ucsc.edu/~haber/ph116A/adjugate.pdf)</sup> |
| Rank behavior | \( \operatorname{rank}(\operatorname{adj} A) = n \) if \( \operatorname{rank} A = n \); 1 if \( \operatorname{rank} A = n-1 \); 0 if \( \operatorname{rank} A \le n-2 \)<sup>[3](https://nhigham.com/2020/06/16/what-is-the-adjugate-of-a-matrix/)</sup> |
| Direct computation cost | \( n^2 \) determinants of order \( n-1 \), an \( O(n^4) \) process<sup>[1](https://www.sciencedirect.com/science/article/pii/S0024379598100988)</sup> |
| Other names | Classical adjoint, adjunct matrix; sometimes called the adjoint matrix in older literature<sup>[4](https://reference.wolfram.com/language/ref/Adjugate.html.en?source=footer)</sup><sup> • </sup><sup>[5](https://sep.stanford.edu/sep/prof/gee/ajt/paper_html/node21.html)</sup>  |

## Definition and basic identity

For an \( n \times n \) matrix \( A = [a_{ij}] \), the minor \( M_{ij} \) is the determinant of the \( (n-1) \times (n-1) \) matrix left after deleting row \( i \) and column \( j \), and the cofactor is \( C_{ij} = (-1)^{i+j} M_{ij} \).<sup>[2](https://scipp-legacy.pbsci.ucsc.edu/~haber/ph116A/adjugate.pdf)</sup> The adjugate is \( \operatorname{adj} A = C^T \), so its \( (i,j) \) entry is \( C_{ji} \); in minor notation, \( (\operatorname{adj} A)_{ij} = (-1)^{i+j} M_{ji}(A) \).<sup>[2](https://scipp-legacy.pbsci.ucsc.edu/~haber/ph116A/adjugate.pdf)</sup><sup> • </sup><sup>[6](https://planetmath.org/Adjugate)</sup>  The transposition is what turns the cofactor expansion of the determinant into a matrix identity: combining the expansion along a row with the fact that a matrix with two equal rows has determinant zero gives \( \delta_{ik} \det A = \sum_j a_{kj} C_{ij} \), which in matrix form is \( A\,(\operatorname{adj} A) = (\det A)\,I \), and the same argument on columns gives \( (\operatorname{adj} A)\,A = (\det A)\,I \).<sup>[2](https://scipp-legacy.pbsci.ucsc.edu/~haber/ph116A/adjugate.pdf)</sup>

For a 2 × 2 matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \), the adjugate is \( \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \).<sup>[4](https://reference.wolfram.com/language/ref/Adjugate.html.en?source=footer)</sup> For a 3 × 3 matrix the adjugate method is a standard route to the inverse in coursework: compute the determinant, form the nine cofactors, transpose, and divide by the determinant.<sup>[7](https://sph.umich.edu/biostat/biostart/matrix_inverse.html)</sup>

## Relation to the inverse

When \( \det A \neq 0 \), dividing the defining identity by the determinant gives \( A^{-1} = \operatorname{adj} A / \det A \).<sup>[2](https://scipp-legacy.pbsci.ucsc.edu/~haber/ph116A/adjugate.pdf)</sup> Equivalently, for an invertible matrix the adjugate can be defined as \( \operatorname{adj}(A) = \det(A)\,A^{-1} \).<sup>[6](https://planetmath.org/Adjugate)</sup> The adjugate itself can be computed without ever using division, so it remains available where an inverse does not exist, and it can be useful in applications involving singular matrices.<sup>[5](https://sep.stanford.edu/sep/prof/gee/ajt/paper_html/node21.html)</sup>

**Behavior at singularity.** The rank of the adjugate falls in a fixed pattern. If \( \operatorname{rank} A = n \), the adjugate is invertible with rank \( n \). If \( \operatorname{rank} A = n-1 \), then \( \det A = 0 \), so \( A\,(\operatorname{adj} A) = 0 \), meaning every column of the adjugate lies in the null space of \( A \); since that null space is one dimensional and at least one \( (n-1) \times (n-1) \) minor is nonzero, \( \operatorname{rank}(\operatorname{adj} A) = 1 \), and \( \det(\operatorname{adj} A) = 0 \) for \( n \ge 2 \). If \( \operatorname{rank} A \le n-2 \), every \( (n-1) \times (n-1) \) submatrix is singular, so \( \operatorname{adj}(A) \) is the zero matrix.<sup>[3](https://nhigham.com/2020/06/16/what-is-the-adjugate-of-a-matrix/)</sup>

## Numerical computation

Computing the adjugate from its definition requires \( n^2 \) determinants of order \( n-1 \), a prohibitively expensive \( O(n^4) \) process.<sup>[1](https://www.sciencedirect.com/science/article/pii/S0024379598100988)</sup> The alternative formula \( \operatorname{adj}(A) = \det(A)\,A^{-1} \) breaks down when \( A \) is singular and is potentially unstable when \( A \) is ill-conditioned with respect to inversion.<sup>[1](https://www.sciencedirect.com/science/article/pii/S0024379598100988)</sup> G. W. Stewart, a numerical analyst known for work in matrix computations,<sup>[8](https://books.google.com/books/about/Matrix_Algorithms.html?id=RfLOO2_VM04C)</sup> showed in a 1998 paper in *Linear Algebra and its Applications* that the adjugate can be perfectly conditioned even when \( A \) is ill-conditioned, and that with due care it can be accurately computed from the inverse even when the inverse itself has been inaccurately computed.<sup>[1](https://www.sciencedirect.com/science/article/pii/S0024379598100988)</sup>

For rank \( n-1 \) matrices, a practical route uses the singular value decomposition (SVD). If \( \sigma_1 \ge \cdots \ge \sigma_n \) are the singular values, then

\[ \operatorname{adj}(A) = \rho\, \sigma_1 \sigma_2 \cdots \sigma_{n-1}\, v_n u_n^{*}, \]

where \( \rho = \overline{\det(V)}\,\det(U) \) has modulus 1 and \( v_n \), \( u_n \) are the last columns of \( V \) and \( U \).<sup>[3](https://nhigham.com/2020/06/16/what-is-the-adjugate-of-a-matrix/)</sup> Nicholas J. Higham, a numerical analyst at the [University of Manchester](https://www.edgepedia.org/university-of-manchester), notes that the definition by cofactors is not a good computational means and gives a MATLAB function based on this SVD formula.<sup>[3](https://nhigham.com/2020/06/16/what-is-the-adjugate-of-a-matrix/)</sup>

More broadly, the cofactor and adjugate formulae are mainly of theoretical interest: they are useful in proofs, but for numerical work they become impractical once \( n \) grows beyond about 4, and row reduction is the preferred method for determinants and inverses.<sup>[2](https://scipp-legacy.pbsci.ucsc.edu/~haber/ph116A/adjugate.pdf)</sup>

## Applications

**Cramer's rule.** The adjugate underlies [Cramer's rule](https://www.edgepedia.org/cramers-rule), which solves an \( n \times n \) system \( A\mathbf{x} = \mathbf{b} \) by ratios of determinants rather than row operations or an explicit inverse.<sup>[9](https://learnmath.org/topics/college-algebra/cramers-rule/)</sup> Analogues of Cramer's rule built on adjoint-matrix analogs give determinantal representations for least squares solutions with minimum norm and for [Drazin inverse](https://www.edgepedia.org/drazin-inverse) solutions of singular linear systems.<sup>[10](https://arxiv.org/abs/1503.06997)</sup>

**Eigenvalues and the Jordan form.** Thompson and McEnteggert's theorem states that for a [Hermitian matrix](https://www.edgepedia.org/hermitian-matrix) \( A \) and a simple eigenvalue \( \lambda_i \) with unit eigenvector \( z_i \), \( \operatorname{Adj}(\lambda_i I - A) \) is a rescaling of the orthogonal projection onto the eigenspace of \( \lambda_i \).<sup>[11](https://arxiv.org/abs/2303.09953)</sup> Recent work extends this to repeated eigenvalues and general matrices, factoring higher-order derivatives of the adjugate through the nilpotent and projector matrices of the Jordan decomposition.<sup>[11](https://arxiv.org/abs/2303.09953)</sup>

**Geometry and engineering.** The adjugate of the characteristic equation of an eigenvalue problem has been used to study the geometry of the quaternion corresponding to a rotation matrix, showing that the quaternion solution forms a manifold with several single-valued sectors represented by the adjugate, with applications to 2D and 3D point-cloud matching and pose estimation.<sup>[12](https://arxiv.org/abs/2205.09116)</sup> Over commutative rings, the classical adjoint is used in studies of generalized invertibility, including a proof of von Neumann's 1936 observation that every matrix over a regular ring is regular and a condition for the existence of the [Moore–Penrose inverse](https://www.edgepedia.org/moore-penrose-inverse).<sup>[13](https://www.researchgate.net/publication/242986811_The_classical_adjoint)</sup>

## Terminology and history

The word "adjoint" carries several meanings in mathematics: the Hilbert adjoint common in physics and engineering, and, in linear algebra, the adjugate matrix, whose elements are signed cofactors.<sup>[5](https://sep.stanford.edu/sep/prof/gee/ajt/paper_html/node21.html)</sup> The adjugate is also known as the classical adjoint or the adjunct matrix, and the name "classical adjoint" distinguishes it from the conjugate transpose operation usually meant by "adjoint".<sup>[4](https://reference.wolfram.com/language/ref/Adjugate.html.en?source=footer)</sup><sup> • </sup><sup>[6](https://planetmath.org/Adjugate)</sup>  The adjugate matrix is sometimes called the adjoint matrix, particularly in the older literature.<sup>[5](https://sep.stanford.edu/sep/prof/gee/ajt/paper_html/node21.html)</sup>

The underlying theory goes back to [Augustin-Louis Cauchy](https://www.edgepedia.org/augustin-louis-cauchy)'s 1812 memoir, the first use of "determinant" in its modern sense, in which he reproved earlier results and gave new results on minors and adjoints; the multiplication theorem for determinants was proved there for the first time.<sup>[14](https://mathshistory.st-andrews.ac.uk/HistTopics/Matrices_and_determinants/)</sup> Thomas Muir's history records Cauchy's 1812 theorem regarding the adjugate, and Jacobi's theorem on any minor of the adjugate, as early landmarks of compound determinants.<sup>[15](https://www.cambridge.org/core/journals/proceedings-of-the-royal-society-of-edinburgh/article/abs/xthe-theory-of-compound-determinants-in-the-historical-order-of-its-development-up-to-1860/FE56563282013C6117FA652F3A2FC9C1)</sup> [Arthur Cayley](https://www.edgepedia.org/arthur-cayley)'s 1858 "Memoir on the theory of matrices" later gave an explicit construction of the inverse of a matrix in terms of its determinant.<sup>[14](https://mathshistory.st-andrews.ac.uk/HistTopics/Matrices_and_determinants/)</sup>

## References

1. [G. W. Stewart (1998). On the adjugate matrix. Linear Algebra and its Applications 283, 151–164.](https://www.sciencedirect.com/science/article/pii/S0024379598100988)
2. [Howard Haber. Determinant and the Adjugate, Physics 116A notes, UC Santa Cruz.](https://scipp-legacy.pbsci.ucsc.edu/~haber/ph116A/adjugate.pdf)
3. [Nicholas J. Higham (2020). What Is the Adjugate of a Matrix?](https://nhigham.com/2020/06/16/what-is-the-adjugate-of-a-matrix/)
4. [Adjugate: Classical adjoint of a matrix, Wolfram Language Documentation.](https://reference.wolfram.com/language/ref/Adjugate.html.en?source=footer)
5. [Jon Claerbout. The word "adjoint", Stanford Exploration Project.](https://sep.stanford.edu/sep/prof/gee/ajt/paper_html/node21.html)
6. [Adjugate, PlanetMath.](https://planetmath.org/Adjugate)
7. [Linear Algebra: Matrix Inverses and Properties, University of Michigan Biostatistics.](https://sph.umich.edu/biostat/biostart/matrix_inverse.html)
8. [G. W. Stewart. Matrix Algorithms: Volume 1: Basic Decompositions, SIAM.](https://books.google.com/books/about/Matrix_Algorithms.html?id=RfLOO2_VM04C)
9. [Cramer's Rule, learnmath.org.](https://learnmath.org/topics/college-algebra/cramers-rule/)
10. [Cramer's Rule for Generalized Inverse Solutions of Some Matrices Equations, arXiv.](https://arxiv.org/abs/1503.06997)
11. [Higher order derivatives of the adjugate matrix and the Jordan form, arXiv.](https://arxiv.org/abs/2303.09953)
12. [Andrew J. Hanson, Sonya M. Hanson (2022). Exploring the Adjugate Matrix Approach to Quaternion Pose Extraction, arXiv.](https://arxiv.org/abs/2205.09116)
13. [The classical adjoint, ResearchGate.](https://www.researchgate.net/publication/242986811_The_classical_adjoint)
14. [Matrices and determinants, MacTutor History of Mathematics.](https://mathshistory.st-andrews.ac.uk/HistTopics/Matrices_and_determinants/)
15. [Thomas Muir. The Theory of Compound Determinants in the Historical Order of its Development up to 1860, Proceedings of the Royal Society of Edinburgh 28 (1908).](https://www.cambridge.org/core/journals/proceedings-of-the-royal-society-of-edinburgh/article/abs/xthe-theory-of-compound-determinants-in-the-historical-order-of-its-development-up-to-1860/FE56563282013C6117FA652F3A2FC9C1)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Linear and multilinear algebra › Matrix theory › Determinants and minors*

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