adjugate

The adjugate, $\adjA$ , of an $n\times n$ matrix $A$ , is the $n\times n$ matrix

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'' which denotes the conjugate transpose operation.

An equivalent characterization of the adjugate is the following:

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 invertible, the adjugate can be defined as$$ \adjA = \det(A)A^{-1}$$

Yet another definition of the adjugate is the following:

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$$p(t) = \det(t I - A) = t^n - p_1(A) t^{n-1} + \cdots + (-1)^n p_n(A)$$ The equivalence of (2) and (3) follows from the Cayley-Hamilton theorem. The latter states that $p(A)=0$ , which in turn implies that$$A ( A^{n-1} - p_1(A) A^{n-2} + \cdots + (-1)^{n-1} p_{n-1}(A) ) = (-1)^{n-1} \det(A) $$

The adjugate operation enjoys a number of notable properties: