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The adjugate, $\adjA$ , of an $n\times n$ matrix $A$ , is the $n\times n$ matrix \begin{equation} \label{eq:def1} \adjA_{ij} = (-1)^{i+j}\, M_{\!ji}(A) \end{equation}where $M_{\!ji}(A)$ is the indicated minor of $A$ (the determinant obtained by deleting row $j$ and column $i$ from $A$ ). The adjugate is also known as the classical adjoint, to distinguish it from the usual usage of ``adjoint'' which denotes the conjugate transpose operation.
An equivalent characterization of the adjugate is the following: \begin{equation} \label{eq:def2} \adjA A = \det(A) I. \end{equation}The equivalence of (![[*]](http://images.planetmath.org:8080/cache/objects/3604/js//usr/share/latex2html/icons/crossref.png "[*]") ) and () follows easily from the multi-linearity properties of the determinant. Thus, the adjugate operation is closely related to the matrix inverse. Indeed, if $A$ is invertible, the adjugate can be defined as$$ \adjA = \det(A)A^{-1}$$
Yet another definition of the adjugate is the following:
where $p_1(A)=\operatorname{tr}(A), p_2(A),\ldots, p_n(A) = \det(A)$ are the elementary invariant polynomials of $A$ . The latter arise as coefficients in the characteristic polynomial $p(t)$ of $A$ , namely$$p(t) = \det(t I - A) = t^n - p_1(A) t^{n-1} + \cdots + (-1)^n p_n(A)$$ The equivalence of () and () follows from the
Cayley-Hamilton theorem. The latter states that $p(A)=0$ , which in turn implies that$$A ( A^{n-1} - p_1(A) A^{n-2} + \cdots + (-1)^{n-1} p_{n-1}(A) ) = (-1)^{n-1} \det(A) $$
The adjugate operation enjoys a number of notable properties:
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"adjugate" is owned by rmilson. [ full author list (4) | owner history (2) ]
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| Other names: |
classical adjoint |
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Cross-references: properties, number, implies, Cayley-Hamilton theorem, characteristic polynomial, coefficients, invariant polynomials, invertible, matrix inverse, equivalence, characterization, equivalent, operation, conjugate transpose, column, row, determinant, minor, matrix
There are 6 references to this entry.
This is version 14 of adjugate, born on 2002-11-17, modified 2006-09-07.
Object id is 3604, canonical name is MatrixAdjoint.
Accessed 24244 times total.
Classification:
| AMS MSC: | 15A09 (Linear and multilinear algebra; matrix theory :: Matrix inversion, generalized inverses) |
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Pending Errata and Addenda
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