adjugate

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

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

(1)

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” (http://planetmath.org/AdjointEndomorphism) which denotes the conjugate transpose operation.

An equivalent characterization of the adjugate is the following:

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

(2)

The equivalence of (1) and (2) follows easily from the multi-linearity properties (http://planetmath.org/DeterminantAsAMultilinearMapping) 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

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

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 $A$ . The latter arise as coefficients in the characteristic polynomial $p(t)$ of $A$ , namely

$p(t)=\det(tI-A)=t^{n}-p_{1}(A)t^{n-1}+\cdots+(-1)^{n}p_{n}(A).$

The equivalence of (2) and (3) follows from the Cayley-Hamilton theorem. The latter states that $p(A)=0$ , which in turn implies that

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

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)

Title	adjugate
Canonical name	Adjugate
Date of creation	2013-03-22 13:09:42
Last modified on	2013-03-22 13:09:42
Owner	rmilson (146)
Last modified by	rmilson (146)
Numerical id	17
Author	rmilson (146)
Entry type	Definition
Classification	msc 15A09
Synonym	classical adjoint