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== Definition ==
Let {{math|''A''}} be an {{math|''m'' × ''n''}} matrix with [[real number|real]] or [[complex number|complex]] entries.{{efn|The definition, and the purely algebraic part of the theory, of compound matrices requires only that the matrix have entries in a [[commutative ring]]. In this case, the matrix corresponds to a [[module homomorphism|homomorphism]] of [[finitely generated module|finitely generated]] [[free module]]s.}} If {{math|''I''}} is a [[subset]] of size {{math|''r''}} of {{math|{1, ..., ''m''<nowiki>}</nowiki>}} and {{math|''J''}} is a subset of size {{math|''s''}} of {{math|{1, ..., ''n''<nowiki>}</nowiki>}}, then the '''{{math|(''I'', ''J'' )}}-submatrix of {{math|''A''}}''', written {{math|''A''<sub>''I'', ''J''</sub>}} , is the submatrix formed from {{math|''A''}} by retaining only those rows indexed by {{math|''I''}} and those columns indexed by {{math|''J''}}. If {{math|1=''r'' = ''s''}}, then {{math|det ''A''<sub>''I'', ''J''</sub>}} is the '''{{math|(''I'', ''J'' )}}-[[minor (linear algebra)|minor]]''' of {{math|''A''}}.
The '''''r'' th compound matrix''' of {{math|''A''}} is a matrix, denoted {{math|''C''<sub>''r'' </sub>(''A'')}}, is defined as follows. If {{math|''r'' > min(''m'', ''n'')}}, then {{math|''C''<sub>''r'' </sub>(''A'')}} is the unique {{math|0 × 0}} matrix. Otherwise, {{math|''C''<sub>''r'' </sub>(''A'')}} has size <math display="inline">\binom{m}{r} \!\times\! \binom{n}{r}</math>. Its rows and columns are indexed by {{math|''r''}}-element subsets of {{math|{1, ..., ''m''<nowiki>}</nowiki>}} and {{math|{1, ..., ''n''<nowiki>}</nowiki>}}, respectively, in their lexicographic order. The entry corresponding to subsets {{math|''I''}} and {{math|''J''}} is the minor {{math|det ''A''<sub>''I'', ''J''</sub>}}.
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