Editing Adjugate matrix (section)

== Higher adjugates ==
Let {{math|'''A'''}} be an {{math|''n'' × ''n''}} matrix, and fix {{math|''r'' &ge; 0}}.  The '''{{math|''r''}}th higher adjugate''' of {{math|'''A'''}} is an <math display="inline">\binom{n}{r} \!\times\! \binom{n}{r}</math> matrix, denoted {{math|adj<sub>''r''</sub> '''A'''}}, whose entries are indexed by size {{math|''r''}} [[subset]]s {{math|''I''}} and {{math|''J''}} of {{math|{1, ..., ''m''<nowiki>}</nowiki>}} {{Citation needed|date=November 2023}}.  Let {{math|''I''{{i sup|c}}}} and {{math|''J''{{i sup|c}}}} denote the [[complement (set theory)|complements]] of {{math|''I''}} and {{math|''J''}}, respectively.  Also let <math>\mathbf{A}_{I^c, J^c}</math> denote the submatrix of {{math|'''A'''}} containing those rows and columns whose indices are in {{math|''I''{{i sup|c}}}} and {{math|''J''{{i sup|c}}}}, respectively.  Then the {{math|(''I'', ''J'')}} entry of {{math|adj<sub>''r''</sub> '''A'''}} is
:<math>(-1)^{\sigma(I) + \sigma(J)}\det \mathbf{A}_{J^c, I^c},</math>
where {{math|σ(''I'')}} and {{math|σ(''J'')}} are the sum of the elements of {{math|''I''}} and {{math|''J''}}, respectively.

Basic properties of higher adjugates include {{Citation needed|date=November 2023}}:
* {{math|1=adj<sub>0</sub>('''A''') = det '''A'''}}.
* {{math|1=adj<sub>1</sub>('''A''') = adj '''A'''}}.
* {{math|1=adj<sub>''n''</sub>('''A''') = 1}}.
* {{math|1=adj<sub>''r''</sub>('''BA''') = adj<sub>''r''</sub>('''A''') adj<sub>''r''</sub>('''B''')}}.
* <math>\operatorname{adj}_r(\mathbf{A})C_r(\mathbf{A}) = C_r(\mathbf{A})\operatorname{adj}_r(\mathbf{A}) = (\det \mathbf{A})I_{\binom{n}{r}}</math>, where {{math|''C''<sub>''r''</sub>('''A''')}} denotes the {{math|''r''}}th [[compound matrix]].

Higher adjugates may be defined in abstract algebraic terms in a similar fashion to the usual adjugate, substituting <math>\wedge^r V</math> and <math>\wedge^{n-r} V</math> for <math>V</math> and <math>\wedge^{n-1} V</math>, respectively.