Editing Invertible matrix (section)

== Definition ==
An {{mvar|n}}-by-{{mvar|n}} [[square matrix]] {{math|'''A'''}} is called '''invertible''' if there exists an {{mvar|n}}-by-{{mvar|n}} square matrix {{math|'''B'''}} such that<math display="block">\mathbf{AB} = \mathbf{BA} = \mathbf{I}_n ,</math>where {{math|'''I'''<sub>''n''</sub>}} denotes the {{mvar|n}}-by-{{mvar|n}} [[identity matrix]] and the multiplication used is ordinary [[matrix multiplication]].<ref>{{cite book | last=Axler | first=Sheldon | title=Linear Algebra Done Right | volume= | pages=296 | publication-date=2015 | series=[[Undergraduate Texts in Mathematics]] | date=18 December 2014 | edition=3rd | publisher=[[Springer Publishing]] | isbn=978-3-319-11079-0 | author-link=Sheldon Axler}}</ref> If this is the case, then the matrix {{math|'''B'''}} is uniquely determined by {{math|'''A'''}}, and is called the [[Multiplicative inverse|(multiplicative) '''''inverse''''']] of {{math|'''A'''}}, denoted by {{math|'''A'''<sup>−1</sup>}}. '''Matrix inversion''' is the process of finding the matrix which when multiplied by the original matrix gives the identity matrix.<ref>{{cite web|url=https://www.cs.nthu.edu.tw/~jang/book/addenda/matinv/matinv/|author=J.-S. Roger Jang |title=Matrix Inverse in Block Form |date=March 2001}}</ref>

Over a [[field (mathematics)|field]], a square matrix that is ''not'' invertible is called '''singular''' or '''degenerate'''. A square matrix with entries in a field is singular [[if and only if]] its [[determinant]] is zero. Singular matrices are rare in the sense that if a square matrix's entries are randomly selected from any bounded region on the [[number line]] or [[complex plane]], the [[probability]] that the matrix is singular is 0, that is, it will [[almost surely|"almost never"]] be singular. Non-square matrices, i.e. {{mvar|m}}-by-{{mvar|n}} matrices for which {{math|''m'' ≠ ''n''}}, do not have an inverse.  However, in some cases such a matrix may have a [[Inverse element#Matrices|left inverse]] or [[Inverse element#Matrices|right inverse]]. If {{math|'''A'''}} is {{mvar|m}}-by-{{mvar|n}} and the [[rank (linear algebra)|rank]] of {{math|'''A'''}} is equal to {{math|''n''}}, ({{math|''n'' ≤ ''m''}}), then {{math|'''A'''}} has a left inverse, an {{math|''n''}}-by-{{mvar|''m''}} matrix {{math|'''B'''}} such that {{math|1='''BA''' = '''I'''<sub>''n''</sub>}}.  If {{math|'''A'''}} has rank {{math|''m''}} ({{math|''m'' ≤ ''n''}}), then it has a right inverse, an {{mvar|n}}-by-{{mvar|m}} matrix {{math|'''B'''}} such that {{math|1='''AB''' = '''I'''<sub>''m''</sub>}}.

While the most common case is that of matrices over the [[real number|real]] or [[complex number|complex]] numbers, all of those definitions can be given for matrices over any [[algebraic structure]] equipped with [[addition]] and [[multiplication]] (i.e. [[ring (mathematics)|rings]]). However, in the case of a ring being [[Commutative ring|commutative]], the condition for a square matrix to be invertible is that its determinant is invertible in the ring, which in general is a stricter requirement than it being nonzero. For a [[noncommutative ring]], the usual determinant is not defined. The conditions for existence of left-inverse or right-inverse are more complicated, since a notion of rank does not exist over rings.

The set of {{math|''n'' × ''n''}} invertible matrices together with the operation of [[matrix multiplication]] and entries from ring {{mvar|R}} form a [[Group (mathematics)|group]], the [[general linear group]] of degree {{mvar|n}}, denoted {{math|GL<sub>''n''</sub>(''R'')}}.