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Non-square matrices (''m''-by-''n'' matrices for which ''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 '''A''' is ''m''-by-''n'' and the [[rank (linear algebra)|rank]] of '''A''' is equal to ''n'', then '''A''' has a left inverse: an ''n''-by-''m'' matrix '''B''' such that '''BA''' = '''I'''. If '''A''' has rank ''m'', then it has a right inverse: an ''n''-by-''m'' matrix '''B''' such that '''AB''' = '''I'''.
{{anchor|singular}} A square matrix that is not invertible is called '''singular''' or '''degenerate'''. A square matrix is singular [[if and only if]] its [[determinant]] is 0. Singular matrices are rare in the sense that
While the most common case is that of matrices over the [[real number|real]] or [[complex number|complex]] numbers, all these definitions can be given for matrices over any [[ring (mathematics)|commutative ring]]. However, in this case the condition for a square matrix to be invertible is that its determinant is invertible in the ring, which in general is a much stricter requirement than being nonzero. The conditions for existence of left-inverse resp. right-inverse are more complicated since a notion of rank does not exist over rings.
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