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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=0895-7177|doi=10.1016/S0895-7177(01)00058-9|doi-access=free}}</ref><ref name="Price1947">{{cite journal|last1=Price|first1=G. B.|authorlink=G. B. Price|title=Some Identities in the Theory of Determinants|journal=The American Mathematical Monthly|volume=54|issue=2|year=1947|pages=75–90|issn=0002-9890|doi=10.2307/2304856|jstor=2304856}}</ref>
:<math>\operatorname{adj}_r(A) = JC_{n-r}(SAS)^TJ.</math>
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