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{{Inline|date=June 2018}}

In [[linear algebra]], an ''n''-by-''n'' [[square matrix]] '''A''' is called '''invertible''' (also '''nonsingular''' or '''nondegenerate''') if there exists an ''n''-by-''n'' square matrix '''B''' such that

:<math>\mathbf{AB} = \mathbf{BA} = \mathbf{I}_n \  </math>

where '''I'''<sub>''n''</sub> denotes the ''n''-by-''n'' [[identity matrix]] and the multiplication used is ordinary [[matrix multiplication]]. If this is the case, then the matrix '''B''' is uniquely determined by '''A''' and is called the '''''inverse''''' of '''A''', denoted by '''A'''<sup>−1</sup>.

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 a square matrix randomly selected from a [[continuous uniform distribution]] on its entries will [[almost never]] be singular.

Non-square matrices (''m''-by-''n'' matrices for which {{nowrap|''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'' (''n'' ≤ ''m''), then '''A''' has a left inverse: an ''n''-by-''m'' matrix '''B''' such that {{nowrap|1='''BA''' = '''I'''<sub>''n''</sub>}}.  If '''A''' has rank ''m'' (''m'' ≤ ''n''), then it has a right inverse: an ''n''-by-''m'' matrix '''B''' such that {{nowrap|1='''AB''' = '''I'''<sub>''m''</sub>}}.

'''Matrix inversion''' is the process of finding the matrix '''B''' that satisfies the prior equation for a given invertible matrix '''A'''.

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)|ring]]. However, in the case of the ring being 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 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 {{nowrap|''n'' × ''n''}} invertible matrices together with the operation of matrix multiplication form a [[Group (mathematics)|group]], the [[general linear group]] of degree ''n''.

== Properties ==

=== The invertible matrix theorem ===
Let '''A''' be a square ''n'' by ''n'' matrix over a [[field (mathematics)|field]] ''K'' (for example the field '''R''' of real numbers). The following statements are equivalent, that is, for any given matrix they are either all true or all false:

: '''A''' is invertible, that is, '''A''' has an inverse, is nonsingular, or is nondegenerate.
: '''A''' is [[Row equivalence|row-equivalent]] to the ''n''-by-''n'' [[identity matrix]] '''I'''<sub>''n''</sub>.
: '''A''' is [[Row equivalence|column-equivalent]] to the ''n''-by-''n'' [[identity matrix]] '''I'''<sub>''n''</sub>.
: '''A''' has ''n'' [[pivot position]]s.
: {{nowrap|[[determinant|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]] in that ring.
: '''A''' has full rank; that is, {{nowrap|1=[[rank (linear algebra)|rank]] '''A''' = ''n''}}.
: The equation {{nowrap|1='''Ax''' = '''0'''}} has only the trivial solution '''x'''&nbsp;=&nbsp;'''0'''.
: The [[Kernel (linear algebra)|kernel]] of '''A''' is trivial, that is, it contains only the null vector as an element, ''ker('''''A''''')'' = {'''0'''}.
: {{nowrap|1=[[null space|Null]] '''A''' = {0}{{null}}}}.
: The equation '''Ax''' = '''b''' has exactly one solution for each '''b''' in ''K<sup>n</sup>''.
: The columns of '''A''' are [[linear independence|linearly independent]].
: The columns of '''A''' [[linear span|span]] ''K<sup>n</sup>''.
: {{nowrap|1=[[Row and column spaces|Col]] '''A''' = ''K<sup>n</sup>''}}.
: The columns of '''A''' form a [[basis of a vector space|basis]] of ''K<sup>n</sup>''.
: The linear transformation mapping '''x''' to '''Ax'''  is a [[bijection]] from ''K<sup>n</sup>'' to ''K<sup>n</sup>''.
: There is an ''n''-by-''n'' matrix '''B''' such that {{nowrap|1='''AB''' = '''I'''<sub>''n''</sub> = '''BA'''}}.
: The [[transpose]] '''A'''<sup>T</sup> is an invertible matrix (hence rows of '''A''' are [[linear independence|linearly independent]], span ''K<sup>n</sup>'', and form a [[basis of a vector space|basis]] of ''K<sup>n</sup>'').
: The number 0 is not an [[eigenvalue]] of '''A'''.
: The matrix '''A''' can be expressed as a finite product of [[elementary matrix|elementary matrices]].
: The matrix '''A''' has a left inverse (that is, there exists a '''B''' such that {{nowrap|1='''BA''' = '''I'''}}) ''or'' a right inverse (that is, there exists a '''C''' such that {{nowrap|1='''AC''' = '''I'''}}), in which case both left and right inverses exist and {{nowrap|1='''B''' = '''C''' = '''A'''<sup>−1</sup>}}.

=== Other properties ===
Furthermore, the following properties hold for an invertible matrix '''A''':

* ('''A'''<sup>−1</sup>)<sup>−1</sup> = '''A''';
* (''k'''''A''')<sup>−1</sup> = ''k''<sup>−1</sup>'''A'''<sup>−1</sup> for nonzero scalar ''k'';
* ('''Ax''')<sup>+</sup> = '''x'''<sup>+</sup>'''A'''<sup>−1</sup> where <sup>+</sup> denotes the [[Moore–Penrose inverse]] and '''x''' is a vector;
* ('''A'''<sup>T</sup>)<sup>−1</sup> = ('''A'''<sup>−1</sup>)<sup>T</sup>;
* For any invertible ''n''-by-''n'' matrices '''A''' and '''B''', ('''AB''')<sup>−1</sup> = '''B'''<sup>−1</sup>'''A'''<sup>−1</sup>.  More generally, if '''A'''<sub>1</sub>, ..., '''A'''<sub>''k''</sub> are invertible ''n''-by-''n'' matrices, then ('''A'''<sub>1</sub>'''A'''<sub>2</sub>⋅⋅⋅'''A'''<sub>''k''−1</sub>'''A'''<sub>''k''</sub>)<sup>−1</sup> = '''A'''{{su|b=''k''|p=−1}}'''A'''{{su|b=''k''−1|p=−1}}⋯'''A'''{{su|b=2|p=−1}}'''A'''{{su|b=1|p=−1}};
* det '''A'''<sup>−1</sup> = (det '''A''')<sup>−1</sup>.

The rows of the inverse matrix '''V''' of a matrix '''U''' are [[orthonormal]] to the columns of '''U''' (and vice versa interchanging rows for columns). To see this, suppose that '''UV = VU = I''' where we write the rows of '''V''' as <math>v_i^{T}</math> and the columns of '''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^{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 '''U''' are known and then applying the iterative [[Gram–Schmidt process]] to this initial set to determine the rows of the inverse '''V'''. 

A matrix that is its own inverse, that is, such that {{nowrap|1='''A''' = '''A'''<sup>−1</sup>}} and {{nowrap|1='''A'''<sup>2</sup> = '''I'''}}, is called an [[Involutory matrix|involutory matrix]].

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

If <math>A</math> is an <math>n\times n</math> invertible matrix, then
: <math>A^{-1} = \frac{1}{\det(A)} \operatorname{adj}(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 '''A''' and '''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 | postscript=<!--None-->}}.</ref>

=== Density ===
Over the field of real numbers, the set of singular ''n''-by-''n'' matrices, considered as a subset of '''R'''<sup>''n''&times;''n''</sup>, is a [[null set]], that is, has [[Lebesgue measure|Lebesgue]] [[measure zero]]. This is true because singular matrices are the roots of the [[determinant]] function. This is a continuous function because it is a polynomial in the entries of the matrix. Thus in the language of [[measure theory]], [[almost all]] ''n''-by-''n'' matrices are invertible.

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

In practice however, one may encounter non-invertible matrices. And in [[numerical analysis|numerical calculations]], matrices which are invertible, but close to a non-invertible matrix, can still be problematic; such matrices are said to be [[Condition number#Matrices|ill-conditioned]].

== Examples ==
Consider the following ''2''-by-''2'' matrix:
: <math>\mathbf{A} = \begin{pmatrix}-1 & \tfrac{3}{2} \\ 1  & -1\end{pmatrix} .</math>
The matrix <math> \mathbf{A} </math> is invertible. To check this, one can compute that <math> \det \mathbf{A} = -1/2 </math>, which is non-zero. 

As an example of a non-invertible, or singular, matrix, consider the matrix
: <math>\mathbf{B} = \begin{pmatrix} -1 & \tfrac{3}{2} \\ \tfrac{2}{3} & -1 \end{pmatrix} .</math>
The determinant of <math> \mathbf{B} </math> is 0, which is a necessary and sufficient condition for a matrix to be non-invertible.

== Methods of matrix inversion ==

=== Gaussian elimination ===
[[Gaussian elimination#Finding the inverse of a matrix|Gauss–Jordan elimination]] is an [[algorithm]] that can be used to determine whether a given matrix is invertible and to find the inverse. An alternative is the [[LU decomposition]], which generates upper and lower triangular matrices, which are easier to invert.

=== Newton's method ===
A generalization of [[Newton's method]] as used for a [[Multiplicative inverse#Algorithms|multiplicative inverse algorithm]] may be convenient, if it is convenient to find a suitable starting seed:

:<math>X_{k+1} = 2X_k - X_k A X_k.</math>

[[Victor Pan]] and [[John Reif]] have done work that includes ways of generating a starting seed.<ref>{{Citation
 | first = Victor
 | last = Pan
 | first2 = John
 | last2 = Reif
 | title = Efficient Parallel Solution of Linear Systems
 | series = Proceedings of the 17th Annual ACM Symposium on Theory of Computing
 | year = 1985
 | place = Providence
 | publisher = [[Association for Computing Machinery|ACM]]}}</ref><ref>
{{Citation
 | first = Victor
 | last = Pan
 | first2 = John
 | last2 = Reif
 | title = Harvard University Center for Research in Computing Technology Report TR-02-85
 | year = 1985
 | place = Cambridge, MA
 | publisher = [[Aiken Computation Laboratory]]}}</ref> [[Byte magazine]] summarised one of their approaches.<ref>{{cite journal
 | authorlink = Thomas E. Phipps Jr.
 | title = The Inversion of Large Matrices
 | journal = [[Byte Magazine]]
 | volume = 11
 | issue = 4
 | pages = 181–190
 | date = April 1986 }}</ref>

Newton's method is particularly useful when dealing with families of related matrices that behave enough like the sequence manufactured for the homotopy above: sometimes a good starting point for refining an approximation for the new inverse can be the already obtained inverse of a previous matrix that nearly matches the current matrix, for example, the pair of sequences of inverse matrices used in obtaining [[Matrix square root#By Denman–Beavers iteration|matrix square roots by Denman–Beavers iteration]]; this may need more than one pass of the iteration at each new matrix, if they are not close enough together for just one to be enough. Newton's method is also useful for "touch up" corrections to the Gauss–Jordan algorithm which has been contaminated by small errors due to [[Round-off error|imperfect computer arithmetic]].

=== Cayley–Hamilton method ===
The [[Cayley–Hamilton theorem]] allows the inverse of <math>A</math> to be expressed in terms of det(<math>A</math>), traces and powers of <math>A</math>:<ref>A proof can be found in the Appendix B of {{cite journal | last1 = Kondratyuk | first1 = L. A. | last2 = Krivoruchenko | first2 = M. I. | year = 1992 | title = Superconducting quark matter in SU(2) color group | url = https://www.researchgate.net/publication/226920070 | journal = Zeitschrift für Physik A | volume = 344 | issue = | pages = 99–115 | doi = 10.1007/BF01291027 }}</ref>
: <math>\mathbf{A}^{-1} = \frac{1}{\det(\mathbf{A})} \sum_{s=0}^{n-1} \mathbf{A}^s \sum_{k_1,k_2,\ldots,k_{n-1}} \prod_{l=1}^{n-1} \frac{(-1)^{k_l + 1}}{l^{k_l}k_l!} \operatorname{tr}(\mathbf{A}^l)^{k_l},</math>
where <math>n</math> is dimension of <math>A</math>, and <math>\operatorname{tr}(A)</math> is the [[Trace (linear algebra)|trace]] of matrix <math>A</math> given by the sum of the main diagonal.  The sum is taken over <math>s</math> and the sets of all <math>k_l \geq 0</math> satisfying the linear [[Diophantine equation]] 
: <math>s + \sum_{l=1}^{n-1} lk_l = n - 1.</math>
The formula can be rewritten in terms of complete [[Bell polynomials]] of arguments <math>t_l = - (l - 1)! \operatorname{tr}(A^l)</math> as
: <math> 
\mathbf{A}^{-1} = \frac{1}{\det(\mathbf{A})} \sum_{s=1}^n \mathbf{A}^{s-1} \frac{(-1)^{n - 1}}{(n - s)!} B_{n-s}(t_1, t_2, \ldots, t_{n-s}).
</math>

=== Eigendecomposition ===
{{Main article|Eigendecomposition of a matrix}}
If matrix '''A''' can be eigendecomposed, and if none of its eigenvalues are zero, then '''A''' is invertible and its inverse is given by
: <math>\mathbf{A}^{-1} = \mathbf{Q}\mathbf{\Lambda}^{-1}\mathbf{Q}^{-1},</math>
where '''Q''' is the square (''N''×''N'') matrix whose ''i''-th column is the eigenvector <math>q_i</math> of '''A''', and '''Λ''' is the [[diagonal matrix]] whose diagonal elements are the corresponding eigenvalues, that is, <math>\Lambda_{ii} = \lambda_i</math>.
Furthermore, because '''Λ''' is a diagonal matrix, its inverse is easy to calculate:
: <math>\left[\Lambda^{-1}\right]_{ii} = \frac{1}{\lambda_i}.</math>

=== Cholesky decomposition ===
{{Main article|Cholesky decomposition}}
If matrix '''A''' is [[Positive definite matrix|positive definite]], then its inverse can be obtained as
: <math>\mathbf{A}^{-1} = (\mathbf{L}^{*})^{-1} \mathbf{L}^{-1} , </math>
where '''L''' is the lower triangular [[Cholesky decomposition]] of '''A''', and '''L*''' denotes the conjugate transpose of '''L'''.

=== Analytic solution ===
{{Main article|Cramer's rule}}
Writing the transpose of the [[matrix of cofactors]], known as an [[adjugate matrix]], can also be an efficient way to calculate the inverse of ''small'' matrices, but this recursive method is inefficient for large matrices. To determine the inverse, we calculate a matrix of cofactors:

: <math>\mathbf{A}^{-1}={1 \over \begin{vmatrix}\mathbf{A}\end{vmatrix}}\mathbf{C}^{\mathrm{T}}={1 \over \begin{vmatrix}\mathbf{A}\end{vmatrix}}
\begin{pmatrix}
\mathbf{C}_{11} & \mathbf{C}_{21} & \cdots & \mathbf{C}_{n1} \\
\mathbf{C}_{12} & \mathbf{C}_{22} & \cdots & \mathbf{C}_{n2} \\
\vdots & \vdots & \ddots & \vdots \\
\mathbf{C}_{1n} & \mathbf{C}_{2n} & \cdots & \mathbf{C}_{nn} \\
\end{pmatrix}</math>

so that
: <math>\left(\mathbf{A}^{-1}\right)_{ij}={1 \over \begin{vmatrix}\mathbf{A}\end{vmatrix}}\left(\mathbf{C}^{\mathrm{T}}\right)_{ij}={1 \over \begin{vmatrix}\mathbf{A}\end{vmatrix}}\left(\mathbf{C}_{ji}\right)</math>
where |'''A'''| is the [[determinant]] of '''A''', '''C''' is the [[matrix of cofactors]], and '''C'''<sup>T</sup> represents the matrix [[transpose]].

==== Inversion of 2 × 2 matrices ====
The ''cofactor equation'' listed above yields the following result for {{nowrap|2 × 2}} matrices. Inversion of these matrices can be done as follows:<ref>{{cite book
|title=Introduction to linear algebra
|edition=3rd
|first1=Gilbert
|last1=Strang
|publisher=SIAM
|year=2003
|isbn=978-0-9614088-9-3
|page=71
|url=https://books.google.com/books?id=Gv4pCVyoUVYC}}, [https://books.google.com/books?id=Gv4pCVyoUVYC&pg=PA71 Chapter 2, page 71]
</ref>
: <math>\mathbf{A}^{-1} = \begin{bmatrix}
a & b \\ c & d \\ 
\end{bmatrix}^{-1} =
\frac{1}{\det \mathbf{A}} \begin{bmatrix}
\,\,\,d & \!\!-b \\ -c & \,a \\ 
\end{bmatrix} =
\frac{1}{ad - bc} \begin{bmatrix}
\,\,\,d & \!\!-b \\ -c & \,a \\ 
\end{bmatrix}.</math>
This is possible because {{nowrap|1/(''ad'' − ''bc'')}} is the reciprocal of the determinant of the matrix in question, and the same strategy could be used for other matrix sizes.

The Cayley–Hamilton method gives
: <math> \mathbf{A}^{-1}=\frac{1}{\det \mathbf{A}} \left[ \left( \operatorname{tr}\mathbf{A} \right) \mathbf{I} - \mathbf{A} \right] .</math>

==== Inversion of 3 × 3 matrices ====
A computationally efficient {{nowrap|3 × 3}} matrix inversion is given by
: <math>\mathbf{A}^{-1} = \begin{bmatrix}
a & b & c\\ d & e & f \\ g & h & i\\
\end{bmatrix}^{-1} =
\frac{1}{\det(\mathbf{A})} \begin{bmatrix}
\, A & \, B & \,C \\ \, D & \, E & \, F \\ \, G & \, H & \, I\\
\end{bmatrix}^\mathrm{T} =
\frac{1}{\det(\mathbf{A})} \begin{bmatrix}
\, A & \, D & \,G \\ \, B & \, E & \,H \\ \, C & \,F & \, I\\
\end{bmatrix}</math>
(where the scalar ''A'' is not to be confused with the matrix '''A''').
If the determinant is non-zero, the matrix is invertible, with the elements of the intermediary matrix on the right side above given by
: <math>\begin{alignat}{6}
 A &={}&  (ei - fh), &\quad& D &={}& -(bi - ch), &\quad& G &={}&  (bf - ce), \\
 B &={}& -(di - fg), &\quad& E &={}&  (ai - cg), &\quad& H &={}& -(af - cd), \\
 C &={}&  (dh - eg), &\quad& F &={}& -(ah - bg), &\quad& I &={}&  (ae - bd). \\
\end{alignat}</math>
The determinant of '''A''' can be computed by applying the [[rule of Sarrus]] as follows:
: <math>\det(\mathbf{A}) = aA+bB+cC.</math>

The Cayley–Hamilton decomposition gives
: <math>
\mathbf{A}^{-1}=\frac{1}{\det (\mathbf{A})}\left[ \frac{1}{2}\left( (\operatorname{tr}\mathbf{A})^{2}-\operatorname{tr}\mathbf{A}^{2}\right) \mathbf{I} -\mathbf{A}\operatorname{tr}\mathbf{A}+\mathbf{A}^{2}\right]. 
</math>

{{anchor|Inversion of 3×3 matrices based on vector products}}
The general {{nowrap|3 × 3}} inverse can be expressed concisely in terms of the [[cross product]] and [[triple product]]. If a matrix <math>\mathbf{A}=\left[\mathbf{x}_0,\;\mathbf{x}_1,\;\mathbf{x}_2\right]</math> (consisting of three column vectors, <math>\mathbf{x}_0</math>, <math>\mathbf{x}_1</math>, and <math>\mathbf{x}_2</math>) is invertible, its inverse is given by 
: <math>\mathbf{A}^{-1}=\frac{1}{\det(\mathbf A)}\begin{bmatrix}
{(\mathbf{x_1}\times\mathbf{x_2})}^\mathrm{T} \\
{(\mathbf{x_2}\times\mathbf{x_0})}^\mathrm{T} \\
{(\mathbf{x_0}\times\mathbf{x_1})}^\mathrm{T} \\
\end{bmatrix}.</math>
The determinant of A, <math>\det(\mathbf{A})</math>, is equal to the triple product of <math>\mathbf{x_0}</math>, <math>\mathbf{x_1}</math>, and <math>\mathbf{x_2}</math>—the volume of the [[parallelepiped]] formed by the rows or columns: 
: <math>\det(\mathbf{A})=\mathbf{x}_0\cdot(\mathbf{x}_1\times\mathbf{x}_2).</math>
The correctness of the formula can be checked by using cross- and triple-product properties and by noting that for groups, left and right inverses always coincide. Intuitively, because of the cross products, each row of <math>\mathbf{A}^{-1}</math> is orthogonal to the non-corresponding two columns of <math>\mathbf{A}</math> (causing the off-diagonal terms of <math>\mathbf{I}=\mathbf{A}^{-1}\mathbf{A}</math> be zero). Dividing by 
: <math>\det(\mathbf{A})=\mathbf{x}_0\cdot(\mathbf{x}_1\times\mathbf{x}_2)</math>
causes the diagonal elements of <math>\mathbf{I}=\mathbf{A}^{-1}\mathbf{A}</math> to be unity. For example, the first diagonal is:
: <math>1 = \frac{1}{\mathbf{x_0}\cdot(\mathbf{x}_1\times\mathbf{x}_2)} \mathbf{x_0}\cdot(\mathbf{x}_1\times\mathbf{x}_2).</math>

==== Inversion of 4 × 4 matrices ====
With increasing dimension, expressions for the inverse of '''A''' get complicated. For {{nowrap|1=''n'' = 4}}, the Cayley–Hamilton method leads to an expression that is still tractable:
: <math> \mathbf{A}^{-1}=\frac{1}{\det (\mathbf{A})}\left[ \frac{1}{6}\left( (\operatorname{tr}\mathbf{A})^{3}-3\operatorname{tr}\mathbf{A}\operatorname{tr}\mathbf{A}^{2}+2\operatorname{tr}\mathbf{A}^{3}\right) \mathbf{I} -\frac{1}{2}\mathbf{A}\left( (\operatorname{tr}\mathbf{A})^{2}-\operatorname{tr}\mathbf{A}^{2}\right) +\mathbf{A}^{2}\operatorname{tr}\mathbf{A}-\mathbf{A}^{3}\right]. </math>

=== Blockwise inversion ===
Matrices can also be ''inverted blockwise'' by using the following analytic inversion formula:
{{NumBlk|
: | <math>
\begin{bmatrix} \mathbf{A} & \mathbf{B} \\ \mathbf{C} & \mathbf{D} \end{bmatrix}^{-1} = \begin{bmatrix} \mathbf{A}^{-1}+\mathbf{A}^{-1}\mathbf{B}(\mathbf{D}-\mathbf{CA}^{-1}\mathbf{B})^{-1}\mathbf{CA}^{-1} & -\mathbf{A}^{-1}\mathbf{B}(\mathbf{D}-\mathbf{CA}^{-1}\mathbf{B})^{-1} \\ -(\mathbf{D}-\mathbf{CA}^{-1}\mathbf{B})^{-1}\mathbf{CA}^{-1} & (\mathbf{D}-\mathbf{CA}^{-1}\mathbf{B})^{-1} \end{bmatrix},
</math>
| {{EquationRef|1}} 
}}
where '''A''', '''B''', '''C''' and '''D''' are [[block matrix|matrix sub-blocks]] of arbitrary size. ('''A''' must be square, so that it can be inverted. Furthermore, '''A''' and {{math|'''D''' &minus; '''CA'''<sup>&minus;1</sup>'''B'''}} must be nonsingular.<ref>
{{cite book
  | last = Bernstein
  | first = Dennis
  | title = Matrix Mathematics
  | publisher = Princeton University Press
  | year = 2005
  | pages = 44
  | isbn = 978-0-691-11802-4 }}
</ref>) This strategy is particularly advantageous if '''A''' is diagonal and {{math|'''D''' &minus; '''CA'''<sup>&minus;1</sup>'''B'''}} (the [[Schur complement]] of '''A''') is a small matrix, since they are the only matrices requiring inversion.

This technique was reinvented several times and is due to [[Hans Boltz]] (1923),{{Citation needed|date=December 2009}}<!--Surely this was already known by the end of the 19th century.--> who used it for the inversion of [[geodesy|geodetic]] matrices, and [[Tadeusz Banachiewicz]] (1937), who generalized it and proved its correctness.

The [[nullity theorem]] says that the nullity of '''A''' equals the nullity of the sub-block in the lower right of the inverse matrix, and that the nullity of '''B''' equals the nullity of the sub-block in the upper right of the inverse matrix.

The inversion procedure that led to Equation ({{EquationNote|1}}) performed matrix block operations that operated on '''C''' and '''D''' first. Instead, if '''A''' and '''B''' are operated on first, and provided '''D''' and {{math|'''A''' &minus; '''BD'''<sup>&minus;1</sup>'''C'''}} are nonsingular,<ref>
{{cite book
  | last = Bernstein
  | first = Dennis
  | title = Matrix Mathematics
  | publisher = Princeton University Press
  | year = 2005
  | pages = 45
  | isbn = 978-0-691-11802-4 }}
</ref> the result is
{{NumBlk|
: | <math>
\begin{bmatrix} \mathbf{A} & \mathbf{B} \\ \mathbf{C} & \mathbf{D} \end{bmatrix}^{-1} = \begin{bmatrix} (\mathbf{A}-\mathbf{BD}^{-1}\mathbf{C})^{-1} & -(\mathbf{A}-\mathbf{BD}^{-1}\mathbf{C})^{-1}\mathbf{BD}^{-1} \\ -\mathbf{D}^{-1}\mathbf{C}(\mathbf{A}-\mathbf{BD}^{-1}\mathbf{C})^{-1} & \quad \mathbf{D}^{-1}+\mathbf{D}^{-1}\mathbf{C}(\mathbf{A}-\mathbf{BD}^{-1}\mathbf{C})^{-1}\mathbf{BD}^{-1}\end{bmatrix}.
</math>
| {{EquationRef|2}}
}}
Equating Equations ({{EquationNote|1}}) and ({{EquationNote|2}}) leads to
{{NumBlk|
: | <math>
(\mathbf{A}-\mathbf{BD}^{-1}\mathbf{C})^{-1} = \mathbf{A}^{-1}+\mathbf{A}^{-1}\mathbf{B}(\mathbf{D}-\mathbf{CA}^{-1}\mathbf{B})^{-1}\mathbf{CA}^{-1}\,
</math>
| {{EquationRef|3}}
}}
: <math>
(\mathbf{A}-\mathbf{BD}^{-1}\mathbf{C})^{-1}\mathbf{BD}^{-1} = \mathbf{A}^{-1}\mathbf{B}(\mathbf{D}-\mathbf{CA}^{-1}\mathbf{B})^{-1}\,
</math>
: <math>
\mathbf{D}^{-1}\mathbf{C}(\mathbf{A}-\mathbf{BD}^{-1}\mathbf{C})^{-1} = (\mathbf{D}-\mathbf{CA}^{-1}\mathbf{B})^{-1}\mathbf{CA}^{-1}\,
</math>
: <math>
\mathbf{D}^{-1}+\mathbf{D}^{-1}\mathbf{C}(\mathbf{A}-\mathbf{BD}^{-1}\mathbf{C})^{-1}\mathbf{BD}^{-1} = (\mathbf{D}-\mathbf{CA}^{-1}\mathbf{B})^{-1}\,
</math>
where Equation ({{EquationNote|3}}) is the [[Woodbury matrix identity]], which is equivalent to the [[binomial inverse theorem]].

Since a blockwise inversion of an {{math|{{var|n}} × {{var|n}}}} matrix requires inversion of two half-sized matrices and 6 multiplications between two half-sized matrices, it can be shown that a [[divide and conquer algorithm]] that uses blockwise inversion to invert a matrix runs with the same time complexity as the matrix multiplication algorithm that is used internally.<ref>T. H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein, ''Introduction to Algorithms'', 3rd ed., MIT Press, Cambridge, MA, 2009, §28.2.</ref> There exist [[Matrix multiplication#Algorithms for efficient matrix multiplication|matrix multiplication algorithms]] with a complexity of {{math|''O''(''n''<sup>2.3727</sup>)}} operations, while the best proven lower bound is {{math|[[Big O notation#Family of Bachmann–Landau notations|Ω]]({{var|n}}{{sup|2}} log {{var|n}})}}.<ref>[[Ran Raz]]. On the complexity of matrix product. In Proceedings of the thirty-fourth annual ACM symposium on Theory of computing. ACM Press, 2002. {{doi|10.1145/509907.509932}}.</ref>

This formula simplifies significantly when the upper right block matrix <math>B</math> is the zero matrix. This formulation is useful when the matrices <math>A</math> and <math>D</math> have relatively simple inverse formulas (or [[Moore–Penrose inverse|pseudo inverses]] in the case where the blocks are not all square. In this special case, the block matrix inversion formula stated in full generality above becomes 

:<math>
\begin{bmatrix} \mathbf{A} & \mathbf{0} \\ \mathbf{C} & \mathbf{D} \end{bmatrix}^{-1} = \begin{bmatrix} \mathbf{A}^{-1} & \mathbf{0} \\ -\mathbf{D}^{-1}\mathbf{CA}^{-1} & \mathbf{D}^{-1} \end{bmatrix}.
</math>

=== By Neumann series ===
If a matrix '''A''' has the property that

: <math>\lim_{n \to \infty} (\mathbf I - \mathbf A)^n = 0</math>

then '''A''' is nonsingular and its inverse may be expressed by a [[Neumann series]]:<ref>
{{cite book
  | last = Stewart
  | first = Gilbert
  | title = Matrix Algorithms: Basic decompositions
  | publisher = SIAM
  | year = 1998
  | pages = 55
  | isbn = 978-0-89871-414-2}}
</ref>

: <math>\mathbf A^{-1} = \sum_{n = 0}^\infty (\mathbf I - \mathbf A)^n.</math>

Truncating the sum results in an "approximate" inverse which may be useful as a [[preconditioner]]. Note that a truncated series can be accelerated exponentially by noting that the Neumann series is a [[geometric sum]]. As such, it satisfies 
: <math>\sum_{n=0}^{2^L-1} (\mathbf I -\mathbf A)^n=\prod_{l=0}^{L-1}(\mathbf I + (\mathbf I -\mathbf A)^{2^l})</math>.
Therefore, only <math>2L-2</math> matrix multiplications are needed to compute <math>2^L</math> terms of the sum.

More generally, if '''A''' is "near" the invertible matrix '''X''' in the sense that
: <math>\lim_{n \to \infty} (\mathbf I - \mathbf X^{-1} \mathbf A)^n = 0 \mathrm{~~or~~} \lim_{n \to \infty} (\mathbf I - \mathbf A \mathbf X^{-1})^n = 0</math>
then '''A''' is nonsingular and its inverse is
: <math>\mathbf A^{-1} = \sum_{n = 0}^\infty \left(\mathbf X^{-1} (\mathbf X - \mathbf A)\right)^n \mathbf X^{-1}~.</math>
If it is also the case that {{nowrap|'''A''' − '''X'''}} has [[rank (linear algebra)|rank]] 1 then this simplifies to
: <math>\mathbf A^{-1} = \mathbf X^{-1} - \frac{\mathbf X^{-1} (\mathbf A - \mathbf X) \mathbf X^{-1}}{1+\operatorname{tr}(\mathbf X^{-1} (\mathbf A - \mathbf X))}~.</math>

=== ''P''-adic approximation ===
{{expand section|date=February 2015}}
If ''A'' is a matrix with integer or rational coefficients and we seek a solution in [[arbitrary-precision arithmetic|arbitrary-precision]] rationals, then a [[p-adic]] approximation method converges to an exact solution in <math>O(n^4 \log^2 n)</math>, assuming standard <math>O(n^3)</math> matrix multiplication is used.<ref>{{cite journal | doi = 10.1016/j.cam.2008.07.044 | volume=225 | title=A p-adic algorithm for computing the inverse of integer matrices | journal=Journal of Computational and Applied Mathematics | pages=320–322| year=2009 | last1=Haramoto | first1=H. | last2=Matsumoto | first2=M. }}</ref> The method relies on solving ''n'' linear systems via Dixon's method of ''p''-adic approximation (each in <math>O(n^3 \log^2 n)</math>) and is available as such in software specialized in arbitrary-precision matrix operations, for example, in IML.<ref>{{cite web|url=https://cs.uwaterloo.ca/~astorjoh/iml.html|title=IML - Integer Matrix Library|author=|date=|website=cs.uwaterloo.ca|accessdate=14 April 2018}}</ref>

== Derivative of the matrix inverse ==
Suppose that the invertible matrix '''A''' depends on a parameter ''t''. Then the derivative of the inverse of '''A''' with respect to ''t'' is given by<ref>{{cite book |first=Jan R. |last=Magnus |first2=Heinz |last2=Neudecker |title=Matrix Differential Calculus : with Applications in Statistics and Econometrics |location=New York |publisher=John Wiley & Sons |edition=Revised |year=1999 |pages=151–152 |isbn=0-471-98633-X }}</ref>
: <math> \frac{\mathrm{d}\mathbf{A}^{-1}}{\mathrm{d}t} = - \mathbf{A}^{-1} \frac{\mathrm{d}\mathbf{A}}{\mathrm{d}t} \mathbf{A}^{-1}. </math>

To derive the above expression for the derivative of the inverse of '''A''', one can differentiate the definition of the matrix inverse <math>\mathbf{A}^{-1}\mathbf{A}=\mathbf{I}</math> and then solve for the inverse of '''A''':
: <math>\frac{\mathrm{d}\mathbf{A}^{-1}\mathbf{A}}{\mathrm{d}t}
=\frac{\mathrm{d}\mathbf{A}^{-1}}{\mathrm{d}t}\mathbf{A}
+\mathbf{A}^{-1}\frac{\mathrm{d}\mathbf{A}}{\mathrm{d}t}
=\frac{\mathrm{d}\mathbf{I}}{\mathrm{d}t}
=\mathbf{0}.</math>
Subtracting <math>\mathbf{A}^{-1}\frac{\mathrm{d}\mathbf{A}}{\mathrm{d}t}</math> from both sides of the above and multiplying on the right by <math>\mathbf{A}^{-1}</math> gives the correct expression for the derivative of the inverse:
: <math> \frac{\mathrm{d}\mathbf{A}^{-1}}{\mathrm{d}t} = - \mathbf{A}^{-1} \frac{\mathrm{d}\mathbf{A}}{\mathrm{d}t} \mathbf{A}^{-1}. </math>
Similarly, if <math>\epsilon</math> is a small number then
: <math>\left(\mathbf{A} + \epsilon\mathbf{X}\right)^{-1}
= \mathbf{A}^{-1}
- \epsilon \mathbf{A}^{-1} \mathbf{X} \mathbf{A}^{-1} + \mathcal{O}(\epsilon^2)\,.</math>

More generally, if

: <math> 
\frac { \mathrm{d}f(\mathbf{A})}{ \mathrm{d}t}=\sum_i g_i (\mathbf{A}) \frac{\mathrm{d}\mathbf{A}}{\mathrm{d}t}h_i (\mathbf{A}),
</math>

then,

: <math> f (\mathbf{A}+\epsilon\mathbf{X})= f (\mathbf{A})+\epsilon\sum_i g_i (\mathbf{A})
\mathbf{X}
h_i (\mathbf{A})+ \mathcal{O}(\epsilon^2).
</math>

Given a positive integer <math>n</math>,

: <math>
\begin{align}
\frac{ \mathrm{d}\mathbf{A}^{n}}{ \mathrm{d}t}&=\sum_{i=1}^n \mathbf{A}^{i-1}\frac{ \mathrm{d}\mathbf{A}}{ \mathrm{d}t}\mathbf{A}^{n-i},\\
\frac{ \mathrm{d}\mathbf{A}^{-n}}{ \mathrm{d}t}&=-\sum_{i=1}^n \mathbf{A}^{-i}\frac{ \mathrm{d}\mathbf{A}}{ \mathrm{d}t}\mathbf{A}^{-(n+1-i)}.
\end{align}
</math>

Therefore,

: <math> 
\begin{align}
(\mathbf{A}+\epsilon \mathbf{X})^{n}&=\mathbf{A}^{n}+\epsilon \sum_{i=1}^n \mathbf{A}^{i-1}\mathbf{X}\mathbf{A}^{n-i}+ \mathcal{O}(\epsilon^2),\\
(\mathbf{A}+\epsilon \mathbf{X})^{-n}&=\mathbf{A}^{-n}-\epsilon \sum_{i=1}^n \mathbf{A}^{-i}\mathbf{X}\mathbf{A}^{-(n+1-i)}+ \mathcal{O}(\epsilon^2).
\end{align}
</math>

== Generalized inverse ==
Some of the properties of inverse matrices are shared by [[generalized inverse]]s (for example, the [[Moore–Penrose inverse]]), which can be defined for any ''m''-by-''n'' matrix.

== Applications ==
For most practical applications, it is ''not'' necessary to invert a matrix to solve a [[system of linear equations]]; however, for a unique solution, it ''is'' necessary that the matrix involved be invertible.

Decomposition techniques like [[LU decomposition]] are much faster than inversion, and various fast algorithms for special classes of linear systems have also been developed.

=== Regression/least squares ===
Although an explicit inverse is not necessary to estimate the vector of unknowns, it is the easiest way to estimate their accuracy, found in the diagonal of a matrix inverse (the posterior covariance matrix of the vector of unknowns). However, faster algorithms to compute only the diagonal entries of a matrix inverse are known in many cases.<ref>{{cite journal |first=Lin |last=Lin |first2=Jianfeng |last2=Lu |first3=Lexing |last3=Ying |first4=Roberto |last4=Car |first5=Weinan |last5=E |title=Fast algorithm for extracting the diagonal of the inverse matrix with application to the electronic structure analysis of metallic systems |journal=Communications in Mathematical Sciences |volume=7 |issue=3 |year=2009 |pages=755–777 |doi=10.4310/CMS.2009.v7.n3.a12 }}</ref>

=== Matrix inverses in real-time simulations ===
Matrix inversion plays a significant role in [[computer graphics]], particularly in [[3D graphics]] rendering and 3D simulations. Examples include screen-to-world ray casting, world-to-subspace-to-world object transformations, and physical simulations.

=== Matrix inverses in MIMO wireless communication ===
Matrix inversion also plays a significant role in the [[MIMO]] (Multiple-Input, Multiple-Output) technology in wireless communications. The MIMO system consists of N transmit and M receive antennas. Unique signals, occupying the same frequency band, are sent via N transmit antennas and are received via M receive antennas. The signal arriving at each receive antenna will be a linear combination of the N transmitted signals forming a NxM transmission matrix '''H'''. It is crucial for the matrix '''H''' to be invertible for the receiver to be able to figure out the transmitted information.

== See also ==
* [[Binomial inverse theorem]]
* [[LU decomposition]]
* [[Matrix decomposition]]
* [[Matrix square root]]
* [[Pseudoinverse]]
* [[Singular value decomposition]]
* [[Woodbury matrix identity]]

== References ==
{{Reflist}}

== Further reading ==
* {{springer|title=Inversion of a matrix|id=p/i052440}}
* {{Introduction to Algorithms|2|chapter=28.4: Inverting matrices|pages= 755&ndash;760}}
*{{cite book |first=Dennis S. |last=Bernstein |title=Matrix Mathematics: Theory, Facts, and Formulas |location= |publisher=Princeton University Press |edition=2nd |year=2009 |url=https://books.google.com/books?id=jgEiuHlTCYcC&printsec=frontcover |via=[[Google Books]] }}
*{{cite web |first=Kaare Brandt |last=Petersen |first2=Michael Syskind |last2=Pedersen |date=November 15, 2012 |title=The Matrix Cookbook |pages=17–23 |url=https://www2.imm.dtu.dk/pubdb/views/edoc_download.php/3274/pdf/imm3274.pdf#page=17 }}

== External links ==
{{External links cleanup|date=June 2015}}
*{{cite web |first=Grant |last=Sanderson |title=Inverse Matrices, Column Space and Null Space |work=Essence of Linear Algebra |date=August 15, 2016 |url=https://www.youtube.com/watch?v=uQhTuRlWMxw&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab&index=7 |via=[[YouTube]] }}
*{{cite web |first=Gilbert |last=Strang |title=Linear Algebra Lecture on Inverse Matrices |work=[[MIT OpenCourseWare]] |url=https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/lecture-3-multiplication-and-inverse-matrices/ }}
*[http://netlib.org/lapack/ LAPACK]{{snd}} a collection of FORTRAN subroutines for solving dense linear algebra problems
*[http://www.alglib.net/eigen/ ALGLIB]{{snd}} includes a partial port of the LAPACK to C++, C#, Delphi, etc.
*[http://www.emathhelp.net/calculators/linear-algebra/inverse-of-matrix-calculator/ Symbolic Inverse of Matrix Calculator with steps shown]
*[http://www.vias.org/tmdatanaleng/cc_matrix_pseudoinv.html Moore-Penrose Pseudo-Inverse Matrix]
*[http://nm.mathforcollege.com/mws/gen/04sle/mws_gen_sle_bck_system.pdf Inverse of a Matrix Notes]
*{{planetmath reference|title=Derivative of inverse matrix|id=6362}}

{{Linear algebra}}
{{Matrix classes}}

{{DEFAULTSORT:Invertible Matrix}}
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