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→Relation to adjugate matrices: I^c and J_c must be exchanged in the definition of the r^th higher adjugate matrix, as well as in the Jacobi's formula |
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Let {{math|''A''}} be an {{math|''n'' × ''n''}} matrix. Recall that its '''{{math|''r''}}th higher adjugate matrix''' {{math|adj<sub>''r''</sub>(''A'')}} is the <math display="inline">\binom{m}{r} \times \binom{n}{r}</math> matrix whose {{math|(''I'', ''J'')}} entry is
:<math>(-1)^{\sigma(I) + \sigma(J)} \det A_{
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>
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:<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})_{
Adjugates can also be expressed in terms of compounds. Let {{math|''S''}} denote the ''sign matrix'':
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