Editing Invertible matrix (section)

=== Density ===
Over the field of real numbers, the set of singular {{mvar|n}}-by-{{mvar|n}} matrices, considered as a [[subset]] of {{tmath|\mathbb R^{n \times n},}} is a [[null set]], that is, has [[Lebesgue measure]] zero. That is true because singular matrices are the roots of the [[determinant]] function. It is a [[continuous function]] because it is a [[polynomial]] in the entries of the matrix. Thus in the language of [[measure theory]], [[almost all]] {{mvar|n}}-by-{{mvar|n}} matrices are invertible.

Furthermore, the set of {{mvar|n}}-by-{{mvar|n}} invertible matrices is [[open set|open]] and [[dense set|dense]] in the [[topological space]] of all {{mvar|n}}-by-{{mvar|n}} matrices.  Equivalently, the set of singular matrices is [[closed set|closed]] and [[nowhere dense]] in the space of {{mvar|n}}-by-{{mvar|n}} matrices.

In practice, however, non-invertible matrices may be encountered. In [[numerical analysis|numerical calculations]], matrices that are invertible but close to a non-invertible matrix may still be problematic and are said to be [[Condition number#Matrices|ill-conditioned]].