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adjugate

(Definition)

The adjugate, $\operatorname{adj}(A)$ , of an $n\times n$ matrix , is the $n\times n$ matrix

$\displaystyle \operatorname{adj}(A)_{ij} = (-1)^{i+j}\, M_{\!ji}(A)$

(1)

where $M_{\!ji}(A)$ is the indicated minor of

(the determinant obtained by deleting row

from

). 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:

$\displaystyle \operatorname{adj}(A)A = \det(A) I.$

(2)

The equivalence of (1) and (2) follows easily from the multi-linearity properties of the determinant. Thus, the adjugate operation is closely related to the matrix inverse. Indeed, if

is invertible, the adjugate can be defined as

$\displaystyle \operatorname{adj}(A)= \det(A)A^{-1}$

Yet another definition of the adjugate is the following:

$\displaystyle \operatorname{adj}(A)= p_{n-1}(A) I$	$\displaystyle - p_{n-2}(A) A + p_{n-3}(A) A^2 - \cdots$	(3)
	$\displaystyle + (-1)^{n-2}p_1(A) A^{n-2} + (-1)^{n-1}A^{n-1},$

where $p_1(A)=\operatorname{tr}(A), p_2(A),\ldots, p_n(A) = \det(A)$ are the elementary invariant polynomials of

, namely

$\displaystyle p(t) = \det(t I - A) = t^n - p_1(A) t^{n-1} + \cdots + (-1)^n p_n(A).$

The equivalence of (2) and (3) follows from the Cayley-Hamilton theorem. The latter states that

, which in turn implies that

$\displaystyle A ( A^{n-1} - p_1(A) A^{n-2} + \cdots + (-1)^{n-1} p_{n-1}(A) ) = (-1)^{n-1} \det(A) I$

The adjugate operation enjoys a number of notable properties:

	$\displaystyle \operatorname{adj}(AB) =\operatorname{adj}(B)\operatorname{adj}(A),$	(4)
	$\displaystyle \operatorname{adj}(A^t) = \operatorname{adj}(A)^t,$	(5)
	$\displaystyle \det(\operatorname{adj}(A)) = \det(A)^{n-1}.$	(6)

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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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15A09 (Linear and multilinear algebra; matrix theory :: Matrix inversion, generalized inverses)

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