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 Adjugate 2013-03-22 13:09:42 2013-03-22 13:09:42 rmilson (146) rmilson (146) 17 rmilson (146) Definition msc 15A09 classical adjoint