Editing Invertible matrix (section)

== Properties ==
=== Invertible matrix theorem ===
Let {{math|'''A'''}} be a square {{mvar|n}}-by-{{mvar|n}} matrix over a [[field (mathematics)|field]] {{mvar|K}} (e.g., the field {{tmath|\mathbb R}} of real numbers). The following statements are equivalent, i.e., they are either all true or all false for any given matrix:<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Invertible Matrix Theorem|url=https://mathworld.wolfram.com/InvertibleMatrixTheorem.html|access-date=2020-09-08|website=mathworld.wolfram.com|language=en}}</ref>
* {{math|'''A'''}} is invertible, i.e. it has an inverse under matrix multiplication, i.e., there exists a {{math|'''B'''}} such that {{math|1='''AB''' = '''I'''{{sub|''n''}} = '''BA'''}}. (In that statement, "invertible" can equivalently be replaced with "left-invertible" or "right-invertible" in which one-sided inverses are considered.)
* The linear transformation mapping {{math|'''x'''}} to {{math|'''Ax'''}} is invertible, i.e., it has an inverse under function composition. (There, again, "invertible" can equivalently be replaced with either "left-invertible" or "right-invertible".)
* The [[transpose]] {{math|'''A'''<sup>T</sup>}} is an invertible matrix.
* {{math|'''A'''}} is [[Row equivalence|row-equivalent]] to the {{mvar|n}}-by-{{mvar|n}} [[identity matrix]] {{math|'''I'''{{sub|''n''}}}}.
* {{math|'''A'''}} is [[Row equivalence|column-equivalent]] to the {{mvar|n}}-by-{{mvar|n}} identity matrix {{math|'''I'''{{sub|''n''}}}}.
* {{math|'''A'''}} has {{mvar|n}} [[pivot position]]s.
* {{math|'''A'''}} has full [[Rank (linear algebra)|rank]]: {{math|1=rank '''A''' = ''n''}}.
* {{math|'''A'''}} has a trivial [[Kernel (linear algebra)|kernel]]: {{math|1=ker('''A''') = {'''0'''}.}}
* The linear transformation mapping {{math|'''x'''}} to {{math|'''Ax'''}} is bijective; that is, the equation {{math|1='''Ax''' = '''b'''}} has exactly one solution for each {{math|'''b'''}} in {{mvar|K{{sup|n}}}}. (There, "bijective" can equivalently be replaced with "[[injective]]" or "[[surjective]]".)
* The columns of {{math|'''A'''}} form a [[basis of a vector space|basis]] of {{mvar|K{{sup|n}}}}. (In this statement, "basis" can equivalently be replaced with either "linearly independent set" or "spanning set")
* The rows of {{math|'''A'''}} form a basis of {{mvar|K{{sup|n}}}}. (Similarly, here, "basis" can equivalently be replaced with either "linearly independent set" or "spanning set")
* The [[determinant]] of {{math|'''A'''}} is nonzero: {{math|det '''A''' ≠ 0}}. In general, a square matrix over a [[commutative ring]] is invertible if and only if its determinant is a [[Unit (ring theory)|unit]] (i.e. multiplicatively invertible element) of that ring.
* The number 0 is not an [[eigenvalue]] of {{math|'''A'''}}. (More generally, a number <math>\lambda</math> is an eigenvalue of {{math|'''A'''}} if the matrix <math>\mathbf{A}-\lambda \mathbf{I}</math> is singular, where {{math|'''I'''}} is the identity matrix.)
* The matrix {{math|'''A'''}} can be expressed as a finite product of [[Elementary matrix|elementary matrices]].

=== Other properties ===
Furthermore, the following properties hold for an invertible matrix {{math|'''A'''}}:

* <math>(\mathbf A^{-1})^{-1} = \mathbf A</math>
* <math>(k \mathbf A)^{-1} = k^{-1} \mathbf A^{-1}</math> for nonzero scalar {{mvar|k}}
* <math>(\mathbf{Ax})^+ = \mathbf x^+ \mathbf A^{-1}</math> if {{math|'''A'''}} has orthonormal columns, where {{math|{{sup|+}}}} denotes the [[Moore–Penrose inverse]] and {{math|'''x'''}} is a vector
* <math>(\mathbf A^\mathrm{T})^{-1} = (\mathbf A^{-1})^\mathrm{T}</math>
* For any invertible {{mvar|n}}-by-{{mvar|n}} matrices {{math|'''A'''}} and {{math|'''B'''}}, <math>(\mathbf{AB})^{-1} = \mathbf B^{-1} \mathbf A^{-1}.</math> More generally, if <math>\mathbf A_1, \dots, \mathbf A_k</math> are invertible {{mvar|n}}-by-{{mvar|n}} matrices, then <math>(\mathbf A_1 \mathbf A_2 \cdots \mathbf A_{k-1} \mathbf A_k)^{-1} = \mathbf A_k^{-1} \mathbf A_{k-1}^{-1} \cdots \mathbf A_2^{-1} \mathbf A_1^{-1}.</math>
*<math>\det \mathbf A^{-1} = (\det \mathbf A)^{-1}.</math>

The rows of the inverse matrix {{math|'''V'''}} of a matrix {{math|'''U'''}} are [[orthonormal]] to the columns of {{math|'''U'''}} (and vice versa interchanging rows for columns). To see this, suppose that {{math|1='''UV''' = '''VU''' = '''I'''}} where the rows of {{math|'''V'''}} are denoted as <math>v_i^{\mathrm{T}}</math> and the columns of {{math|'''U'''}} as <math>u_j</math> for <math>1 \leq i,j \leq n.</math> Then clearly, the [[Dot product|Euclidean inner product]] of any two <math>v_i^{\mathrm{T}} u_j = \delta_{i,j}.</math> This property can also be useful in constructing the inverse of a square matrix in some instances, where a set of [[orthogonal]] vectors (but not necessarily orthonormal vectors) to the columns of {{math|'''U'''}} are known. In which case, one can apply the iterative [[Gram–Schmidt process]] to this initial set to determine the rows of the inverse {{math|'''V'''}}. 

A matrix that is its own inverse (i.e., a matrix {{math|'''A'''}} such that {{math|1='''A''' = '''A'''{{sup|−1}}}} and consequently {{math|1='''A'''{{sup|2}} = '''I'''}}) is called an [[involutory matrix]].

=== In relation to its adjugate ===
The [[Adjugate matrix|adjugate]] of a matrix {{math|'''A'''}} can be used to find the inverse of {{math|'''A'''}} as follows:

If {{math|'''A'''}} is an invertible matrix, then
: <math>\mathbf{A}^{-1} = \frac{1}{\det(\mathbf{A})} \operatorname{adj}(\mathbf{A}).</math>

=== In relation to the identity matrix ===
It follows from the [[associativity]] of matrix multiplication that if
: <math>\mathbf{AB} = \mathbf{I} \ </math>

for ''finite square'' matrices {{math|'''A'''}} and {{math|'''B'''}}, then also

: <math>\mathbf{BA} = \mathbf{I}\ </math><ref>{{Cite book | last1=Horn | first1=Roger A. | last2=Johnson | first2=Charles R. | title=Matrix Analysis | publisher=[[Cambridge University Press]] | isbn=978-0-521-38632-6 | year=1985 | page=14 }}.</ref>

=== 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]].