The adjugatePlanetmathPlanetmath, adj(A), of an n×n matrix A, is the n×n matrix

adj(A)ij=(-1)i+jMji(A) (1)

where Mji(A) is the indicated minor of A (the determinantMathworldPlanetmath 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 “adjointPlanetmathPlanetmath” ( which denotes the conjugate transposeMathworldPlanetmath operationMathworldPlanetmath.

An equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath characterization of the adjugate is the following:

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 A is invertiblePlanetmathPlanetmath, the adjugate can be defined as


Yet another definition of the adjugate is the following:

adj(A)=pn-1(A)I -pn-2(A)A+pn-3(A)A2- (3)

where p1(A)=tr(A),p2(A),,pn(A)=det(A) are the elementary invariant polynomials of A. The latter arise as coefficients in the characteristic polynomialMathworldPlanetmathPlanetmath p(t) of A, namely


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


The adjugate operation enjoys a number of notable properties:

adj(AB)=adj(B)adj(A), (4)
adj(At)=adj(A)t, (5)
det(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