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where, for any set {{math|''K''}} of integers, {{math|''σ''(''K'')}} is the sum of the elements of {{math|''K''}}. The '''adjugate''' of {{math|''A''}} is its {{math|(''n'' − 1)}}st higher adjugate and is denoted {{math|adj(''A'')}}. The generalized [[Laplace expansion]] formula implies
:<math>C_r(A)\operatorname{adj}_r(A) = \operatorname{adj}_r(A)C_r(A) = (\det A)I_{\binom{n}{r}}.</math>
If {{math|''A''}} is invertible, then
:<math>\operatorname{adj}_r(A^{-1}) = (\det A)^{-1}C_r(A).</math>
A concrete consequence of this is '''Jacobi's formula''' for the minors of an inverse matrix:
:<math>\det(A^{-1})_{I^c, J^c} = (-1)^{\sigma(I) + \sigma(J)} \frac{\det A_{I,J}}{\det A}.</math>
Adjugates can also be expressed in terms of compounds. Let {{math|''S''}} denote the ''sign matrix'':
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and let {{math|''J''}} denote the ''[[exchange matrix]]'':
:<math>J = \begin{pmatrix} & & 1 \\ & \cdots & \\ 1 & & \end{pmatrix}.</math>
Then '''Jacobi's theorem''' states that the {{math|''r''}}th higher adjugate matrix is:<ref name="NambiarSreevalsan2001">{{cite journal|last1=Nambiar|first1=K.K.|last2=Sreevalsan|first2=S.|title=Compound matrices and three celebrated theorems|journal=Mathematical and Computer Modelling|volume=34|issue=3-4|year=2001|pages=251–255|issn=08957177|doi=10.1016/S0895-7177(01)00058-9}}</ref><ref name="Price1947">{{cite journal|last1=Price|first1=G. B.|title=Some Identities in the Theory of Determinants|journal=The American Mathematical Monthly|volume=54|issue=2|year=1947|pages=75|issn=00029890|doi=10.2307/2304856}}</ref>
:<math>\operatorname{adj}_r(A) = JC_{n-r}(SAS)^TJ.</math>
It follows immediately from Jacobi's theorem that
:<math>C_r(A)\, J(C_{n-r}(SAS))^TJ = (\det
Taking adjugates and compounds does not commute. However, compounds of adjugates can be expressed using adjugates of compounds, and vice versa. From the identities
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