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 determinantobtained by deleting row $j$ and column $i$ from $A$ ). The adjugateis also known as the classical adjoint, to distinguish it fromthe usual usage of “adjoint” (http://planetmath.org/AdjointEndomorphism) whichdenotes 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 easilyfrom the multi-linearityproperties (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 ascoefficients in thecharacteristic 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 fromthe Cayley-Hamilton theorem. The latter states that $p(A)=0$ , whichin 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 notableproperties:

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