Editing Unimodular matrix (section)

==Abstract linear algebra==
[[abstract algebra|Abstract linear algebra]] considers matrices with entries from any [[commutative]] [[ring (mathematics)|ring]] <math>R</math>, not limited to the integers.  In this context, a unimodular matrix is one that is invertible over the ring; equivalently, whose determinant is a [[unit (ring theory)|unit]]. This [[group (mathematics)|group]] is denoted <math>\operatorname{GL}_n(R)</math>.<ref>{{cite book |last1=Lang |first1=Serge |title=Algebra |date=2002 |publisher=Springer |edition=rev. 3rd |isbn=0-387-95385-X |page=510, Section XIII.3}}</ref> A rectangular <math>k</math>-by-<math>m</math> matrix is said to be unimodular if it can be extended with <math>m-k</math> rows in <math>R^m</math> to a unimodular square matrix.<ref>
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  | title = Natural Density of Rectangular Unimodular Integer Matrices
  | series = Linear Algebra and its applications
  | publisher = Elsevier
  | pages = 1319–1324
  | volume = 434
  | year = 2011 }}</ref><ref>
{{Citation
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  | first2 = R.
  | title = The density of unimodular matrices over integrally closed subrings of function fields
  | series = Contemporary Developments in Finite Fields and Applications
  | publisher = World Scientific
  | pages = 244–253
  | year = 2016 }}</ref><ref>{{Citation
  | last = Guo
  | first = X.
  | last2 = Yang
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  | title = The probability of rectangular unimodular matrices over Fq [x]
  | series = Linear algebra and its applications
  | publisher = Elsevier
  | pages = 2675–2682
  | year = 2013 }}</ref>

Over a [[field (mathematics)|field]], ''unimodular'' has the same meaning as ''[[invertible matrix|non-singular]]''. ''Unimodular'' here refers to matrices with coefficients in some ring (often the integers) which are invertible over that ring, and one uses ''non-singular'' to mean matrices that are invertible over the field.