TheInfoList

In
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mat ...
, an ''n''-by-''n''
square matrix In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
is called invertible (also nonsingular or nondegenerate), if there exists an ''n''-by-''n'' square matrix such that :$\mathbf = \mathbf = \mathbf_n \$ where denotes the ''n''-by-''n''
identity matrix In linear algebra, the identity matrix of size ''n'' is the ''n'' × ''n'' square matrix In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

and the multiplication used is ordinary
matrix multiplication In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

. If this is the case, then the matrix is uniquely determined by , and is called the (multiplicative) ''inverse'' of , denoted by . Matrix inversion is the process of finding the matrix that satisfies the prior equation for a given invertible matrix . A square matrix that is ''not'' invertible is called singular or degenerate. A square matrix is singular
if and only if In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...
its
determinant In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

is zero. Singular matrices are rare in the sense that if a square matrix's entries are randomly selected from any finite region on the number line or complex plane, the probability that the matrix is singular is 0, that is, it will "almost never" be singular. Non-square matrices (''m''-by-''n'' matrices for which ) do not have an inverse. However, in some cases such a matrix may have a or . If is ''m''-by-''n'' and the
rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking A ranking is a relationship between a set of items such that, for any two items, the first is either "rank ...
of is equal to (), then has a left inverse, an -by- matrix such that . If has rank (), then it has a right inverse, an -by- matrix such that . While the most common case is that of matrices over the
real Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...
or
complex The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London , mottoeng = Let all come who by merit deserve the most reward , established = , type = Public university, Public rese ...

numbers, all these definitions can be given for matrices over any 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 invertible matrices together with the operation of matrix multiplication (and entries from ring ''R'') form a
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
, the
general linear group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
of degree ''n'', denoted .

# Properties

## The invertible matrix theorem

Let A be a square ''n'' by ''n'' matrix over a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
''K'' (e.g., the field R of real numbers). The following statements are equivalent (i.e., they are either all true or all false for any given matrix): : There is an ''n''-by-''n'' matrix B such that . : The matrix A has a left inverse (that is, there exists a B such that ) ''or'' a right inverse (that is, there exists a C such that ), in which case both left and right inverses exist and . : A is invertible, that is, A has an inverse, is nonsingular, or is nondegenerate. : A is row-equivalent to the ''n''-by-''n''
identity matrix In linear algebra, the identity matrix of size ''n'' is the ''n'' × ''n'' square matrix In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

I''n''. : A is column-equivalent to the ''n''-by-''n''
identity matrix In linear algebra, the identity matrix of size ''n'' is the ''n'' × ''n'' square matrix In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

I''n''. : A has ''n''
pivot position The pivot or pivot element is the element of a Matrix (mathematics), matrix, or an array data structure, array, which is selected first by an algorithm (e.g. Gaussian elimination, simplex algorithm, etc.), to do certain calculations. In the case of ...
s. : A has full rank; that is, . : Based on the rank A=n, the equation has only the trivial solution x = 0. and the equation Ax = b has exactly one solution for each b in ''Kn''. : The
kernel Kernel may refer to: Computing * Kernel (operating system) In an operating system with a Abstraction layer, layered architecture, the kernel is the lowest level, has complete control of the hardware and is always in memory. In some systems it ...
of A is trivial, that is, it contains only the null vector as an element, ker(A) = . : The columns of A are
linearly independent In the theory of vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change ...
. : The columns of A span ''Kn''. : . : The columns of A form a
basis Basis may refer to: Finance and accounting *Adjusted basisIn tax accounting, adjusted basis is the net cost of an asset after adjusting for various tax-related items. Adjusted Basis or Adjusted Tax Basis refers to the original cost or other b ...
of ''Kn''. : The linear transformation mapping x to Ax is a
bijection In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

from ''Kn'' to ''Kn''. : . In general, a square matrix over a
commutative ring In , a branch of , a commutative ring is a in which the multiplication operation is . The study of commutative rings is called . Complementarily, is the study of s where multiplication is not required to be commutative. Definition and first e ...
is invertible if and only if its
determinant In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

is a
unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * Unit (album), ...
in that ring. : The number 0 is not an
eigenvalue In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces an ...

of A. : The
transpose In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces an ...

AT is an invertible matrix (hence rows of A are
linearly independent In the theory of vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change ...
, span ''Kn'', and form a
basis Basis may refer to: Finance and accounting *Adjusted basisIn tax accounting, adjusted basis is the net cost of an asset after adjusting for various tax-related items. Adjusted Basis or Adjusted Tax Basis refers to the original cost or other b ...
of ''Kn''). : The matrix A can be expressed as a finite product of elementary matrices.

## Other properties

Furthermore, the following properties hold for an invertible matrix A: * (A−1)−1 = A; * (''k''A)−1 = ''k''−1A−1 for nonzero scalar ''k''; * (Ax)+ = x+A−1 if A has orthonormal columns, where + denotes the
Moore–Penrose inverseIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
and x is a vector; * (AT)−1 = (A−1)T; * For any invertible ''n''-by-''n'' matrices A and B, (AB)−1 = B−1A−1. More generally, if A1, ..., A''k'' are invertible ''n''-by-''n'' matrices, then ; * det A−1 = (det A)−1. The rows of the inverse matrix V of a matrix U are
orthonormalIn linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vector In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quanti ...
to the columns of U (and vice versa interchanging rows for columns). To see this, suppose that UV = VU = I where the rows of V are denoted as $v_i^$ and the columns of U as $u_j$ for $1 \leq i,j \leq n$. Then clearly, the Euclidean inner product of any two $v_i^ u_j = \delta_$. This property can also be useful in constructing the inverse of a square matrix in some instances, where a set of
orthogonal In mathematics, orthogonality is the generalization of the notion of perpendicularity to the linear algebra of bilinear forms. Two elements ''u'' and ''v'' of a vector space with bilinear form ''B'' are orthogonal when . Depending on the bili ...
vectors (but not necessarily orthonormal vectors) to the columns of U are known. In which case, one can apply the iterative
Gram–Schmidt process In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

to this initial set to determine the rows of the inverse V. A matrix that is its own inverse (i.e., a matrix A such that and ), is called an
involutory matrixIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
.

## In relation to its adjugate

The adjugate of a matrix $A$ can be used to find the inverse of $A$ as follows: If $A$ is an invertible matrix, then : $A^ = \frac \left(\operatorname\left(A\right)\right).$

## In relation to the identity matrix

It follows from the associativity of matrix multiplication that if : $\mathbf = \mathbf \$ for ''finite square'' matrices A and B, then also : $\mathbf = \mathbf\$

## Density

Over the field of real numbers, the set of singular ''n''-by-''n'' matrices, considered as a subset of R''n''×''n'', is a
null set In mathematical analysis, a null set N \subset \mathbb is a set that can be covered by a countable union of intervals of arbitrarily small total length. The notion of null set in set theory illustrating the intersection (set theory), interse ...
, that is, has
Lebesgue Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a France, French mathematician known for his Lebesgue integration, theory of integration, which was a generalization of the 17th-century concept of integration—summing the area betwe ...
measure zero In mathematical analysis Analysis is the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calc ...
. This is true because singular matrices are the roots of the
determinant In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

function. This is a continuous function because it is a polynomial in the entries of the matrix. Thus in the language of
measure theory Measure is a fundamental concept of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contai ...
,
almost all In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no genera ...
''n''-by-''n'' matrices are invertible. Furthermore, the ''n''-by-''n'' invertible matrices are a
dense The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its mass Mass is both a property Property (''latin: Res Privata'') in the Abstract and concrete, abstract is what belongs to or ...
open set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
in the
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
of all ''n''-by-''n'' matrices. Equivalently, the set of singular matrices is closed and nowhere dense in the space of ''n''-by-''n'' matrices. In practice however, one may encounter non-invertible matrices. And in numerical calculations, matrices which are invertible, but close to a non-invertible matrix, can still be problematic; such matrices are said to be
ill-conditionedIn the field of numerical analysis, the condition number of a function measures how much the output value of the function can change for a small change in the input argument. This is used to measure how sensitivity analysis, sensitive a function is t ...

.

# Examples

An example with rank of n-1 to be a non-invertible matrix : $\mathbf = \begin 2 & 4\\ 2 & 4 \end .$ We can easily see the rank of this 2*2 matrix is one, which is n-1≠n, so it is a non-invertible matrix. Consider the following 2-by-2 matrix: : $\mathbf = \begin-1 & \tfrac \\ 1 & -1\end .$ The matrix $\mathbf$ is invertible. To check this, one can compute that $\det \mathbf = -\frac$, which is non-zero. As an example of a non-invertible, or singular, matrix, consider the matrix : $\mathbf = \begin -1 & \tfrac \\ \tfrac & -1 \end .$ The determinant of $\mathbf$ is 0, which is a necessary and sufficient condition for a matrix to be non-invertible.

# Methods of matrix inversion

## Gaussian elimination

Gaussian Elimination is the most useful and easiest way to gain the inverse of matrix, so we should explain it carefully with details and examples. Gaussian Elimination is the way used between each row or column, we can use it the change number of the element in matrix just like the way to solve linear equation with two unknown variables. Then, we use this way to get the identity in the right and the change of identity in the left should be the inverse of that matrix. Take an example of matrix: $\mathbf = \begin-1 & \tfrac \\ 1 & -1\end .$ We first exchange it into $\mathbf = \begin-1 & \tfrac & \ 1 & \ 0 \\ 1 & -1\ & 0\ & 1\end .$ Then, we can separate it into two parts, with original matrix in the right side and identity matrix in the left side. We first let row1+row2, and we get $\mathbf = \begin-1 & \tfrac & \ 1 & \ 0 \\ 0 & 1/2 \ & 1\ & 1\end.$ Then let row1-3*row2, which get $\mathbf = \begin-1 & \ 0 & \ -2 & \ -3 \\ 0 & 1/2 \ & 1\ & 1\end.$ Then let row1*-1 and row2*2 to gain the identity matrix in left, then we get the inverse matrix in the right side, which is $\mathbf = \begin1 & \ 0 & \ 2 & \ 3 \\ 0 & 1\ & 2\ & 2\end.$

## Newton's method

A generalization of
Newton's method In numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathem ...

as used for a multiplicative inverse algorithm may be convenient, if it is convenient to find a suitable starting seed: :$X_ = 2X_k - X_k A X_k.$
Victor Pan Victor Yakovlevich Pan (russian: Пан Виктор Яковлевич) is a Soviet The Soviet Union,. officially the Union of Soviet Socialist Republics. (USSR),. was a socialist state A socialist state, socialist republic, or sociali ...
and
John Reif John H. Reif (born 1951) is an American American(s) may refer to: * American, something of, from, or related to the United States of America, commonly known as the United States The United States of America (USA), commonly known as the Uni ...
have done work that includes ways of generating a starting seed.
Byte magazine ''Byte'' (stylized as ''BYTE'') was an American microcomputer computer magazine, magazine, influential in the late 1970s and throughout the 1980s because of its wide-ranging editorial coverage. "''Byte'' magazine, the leading publication serving ...
summarised one of their approaches. 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 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 imperfect computer arithmetic.

## Cayley–Hamilton method

The
Cayley–Hamilton theorem 225px, William Rowan Hamilton (1805–1865), Irish physicist, astronomer, and mathematician, first foreign member of the American National Academy of Sciences. While maintaining opposing position about how geometry should be studied, Hamilton alwa ...
allows the inverse of $A$ to be expressed in terms of $\det\left(A\right)$, traces and powers of $A$: : $\mathbf^ = \frac \sum_^ \mathbf^s \sum_ \prod_^ \frac \operatorname\left\left(\mathbf^l\right\right)^,$ where $n$ is dimension of $A$, and $\operatorname\left(A\right)$ is the
trace Trace may refer to: Arts and entertainment Music * ''Trace'' (Son Volt album), 1995 * ''Trace'' (Died Pretty album), 1993 * Trace (band) Trace was a Netherlands, Dutch progressive rock trio founded by Rick van der Linden in 1974 after leavin ...
of matrix $A$ given by the sum of the main diagonal. The sum is taken over $s$ and the sets of all $k_l \geq 0$ satisfying the linear
Diophantine equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
: $s + \sum_^ lk_l = n - 1.$ The formula can be rewritten in terms of complete
Bell polynomials In combinatorial Combinatorics is an area of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, ...
of arguments $t_l = - \left(l - 1\right)! \operatorname\left\left(A^l\right\right)$ as : $\mathbf^ = \frac \sum_^n \mathbf^ \frac B_\left(t_1, t_2, \ldots, t_\right).$

## Eigendecomposition

If matrix A can be eigendecomposed, and if none of its eigenvalues are zero, then A is invertible and its inverse is given by : $\mathbf^ = \mathbf\mathbf^\mathbf^,$ where $\mathbf$ is the square (''N''×''N'') matrix whose ''i''-th column is the eigenvector $q_i$ of $\mathbf$, and $\mathbf$ is the
diagonal matrix In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and ...
whose diagonal elements are the corresponding eigenvalues, that is, $\Lambda_ = \lambda_i$. If $\mathbf$ is symmetric, $\mathbf$ is guaranteed to be an
orthogonal matrix In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is :Q^\mathrm Q = Q Q^\mathrm = I, where is the transpose In linear algebra, t ...
, therefore $\mathbf^ = \mathbf^\mathrm$. Furthermore, because $\mathbf$ is a diagonal matrix, its inverse is easy to calculate: :

## Cholesky decomposition

If matrix A is
positive definiteIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
, then its inverse can be obtained as : $\mathbf^ = \left\left(\mathbf^*\right\right)^ \mathbf^ ,$ where L is the lower triangular
Cholesky decompositionIn linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and thr ...
of A, and L* denotes the conjugate transpose of L.

## Analytic solution

Writing the transpose of the matrix of cofactors, known as an
adjugate matrixIn linear algebra, the adjugate or classical adjoint of a square matrix In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space ...
, 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: : $\mathbf^ = \mathbf^\mathrm = \begin \mathbf_ & \mathbf_ & \cdots & \mathbf_ \\ \mathbf_ & \mathbf_ & \cdots & \mathbf_ \\ \vdots & \vdots & \ddots & \vdots \\ \mathbf_ & \mathbf_ & \cdots & \mathbf_ \\ \end$ so that : $\left\left(\mathbf^\right\right)_ = \left\left(\mathbf^\right\right)_ = \left\left(\mathbf_\right\right)$ where , A, is the
determinant In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

of A, C is the matrix of cofactors, and CT represents the matrix
transpose In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces an ...

.

### Inversion of 2 × 2 matrices

The ''cofactor equation'' listed above yields the following result for matrices. Inversion of these matrices can be done as follows: : $\mathbf^ = \begin a & b \\ c & d \\ \end^ = \frac \begin \,\,\,d & \!\!-b \\ -c & \,a \\ \end = \frac \begin \,\,\,d & \!\!-b \\ -c & \,a \\ \end.$ This is possible because 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 :

### Inversion of 3 × 3 matrices

A computationally efficient matrix inversion is given by : $\mathbf^ = \begin a & b & c\\ d & e & f \\ g & h & i\\ \end^ = \frac \begin \, A & \, B & \,C \\ \, D & \, E & \, F \\ \, G & \, H & \, I\\ \end^\mathrm = \frac \begin \, A & \, D & \,G \\ \, B & \, E & \,H \\ \, C & \,F & \, I\\ \end$ (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 : $\begin A &=& \left(ei - fh\right), &\quad& D &=& -\left(bi - ch\right), &\quad& G &=& \left(bf - ce\right), \\ B &=& -\left(di - fg\right), &\quad& E &=& \left(ai - cg\right), &\quad& H &=& -\left(af - cd\right), \\ C &=& \left(dh - eg\right), &\quad& F &=& -\left(ah - bg\right), &\quad& I &=& \left(ae - bd\right). \\ \end$ The determinant of A can be computed by applying the rule of Sarrus as follows: : $\det\left(\mathbf\right) = aA + bB + cC.$ The Cayley–Hamilton decomposition gives : The general inverse can be expressed concisely in terms of the
cross product In , the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a on two s in a three-dimensional (named here E), and is denoted by the symbol \times. Given two and , the cross produc ...

and
triple product In geometry and algebra, the triple product is a product of three 3-dimension (vector space), dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, ...
. If a matrix $\mathbf = \begin \mathbf_0 & \mathbf_1 & \mathbf_2\end$ (consisting of three column vectors, $\mathbf_0$, $\mathbf_1$, and $\mathbf_2$) is invertible, its inverse is given by : $\mathbf^ = \frac\begin ^\mathrm \\ ^\mathrm \\ ^\mathrm \end.$ The determinant of A, $\det\left(\mathbf\right)$, is equal to the triple product of $\mathbf$, $\mathbf$, and $\mathbf$—the volume of the
parallelepiped In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of f ...

formed by the rows or columns: : $\det\left(\mathbf\right) = \mathbf_0\cdot\left(\mathbf_1\times\mathbf_2\right).$ 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 $\mathbf^$ is orthogonal to the non-corresponding two columns of $\mathbf$ (causing the off-diagonal terms of $\mathbf = \mathbf^\mathbf$ be zero). Dividing by : $\det\left(\mathbf\right) = \mathbf_0\cdot\left(\mathbf_1\times\mathbf_2\right)$ causes the diagonal elements of $\mathbf=\mathbf^\mathbf$ to be unity. For example, the first diagonal is: : $1 = \frac \mathbf\cdot\left(\mathbf_1\times\mathbf_2\right).$

### Inversion of 4 × 4 matrices

With increasing dimension, expressions for the inverse of A get complicated. For , the Cayley–Hamilton method leads to an expression that is still tractable: :

## Blockwise inversion

Matrices can also be ''inverted blockwise'' by using the following analytic inversion formula: where A, B, C and D are matrix sub-blocks of arbitrary size. (A must be square, so that it can be inverted. Furthermore, A and must be nonsingular.) This strategy is particularly advantageous if A is diagonal and (the
Schur complementIn linear algebra and the theory of matrices Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics) In mathematics, a matrix (plural matrices) is a rectangle, rectangular ''wikt:array, array'' or ''table'' of numbers, sym ...
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), who used it for the inversion of 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 () performed matrix block operations that operated on C and D first. Instead, if A and B are operated on first, and provided D and are nonsingular, the result is Equating Equations () and () leads to where Equation () is the
Woodbury matrix identity In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, which is equivalent to the binomial inverse theorem. If A and D are both invertible, then the above two block matrix inverses can be combined to provide the simple factorization By the
Weinstein–Aronszajn identity In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
, one of the two matrices in the block-diagonal matrix is invertible exactly when the other is. Since a blockwise inversion of an 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 In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of Algorit ...
that uses blockwise inversion to invert a matrix runs with the same time complexity as the
matrix multiplication algorithm Because matrix multiplication is such a central operation in many numerical algorithms, much work has been invested in making matrix multiplication algorithms efficient. Applications of matrix multiplication in computational problems are found in ...
that is used internally. Research into matrix multiplication complexity shows that there exist matrix multiplication algorithms with a complexity of operations, while the best proven lower bound is . This formula simplifies significantly when the upper right block matrix $B$ is the zero matrix. This formulation is useful when the matrices $A$ and $D$ have relatively simple inverse formulas (or 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 :$\begin \mathbf & \mathbf \\ \mathbf & \mathbf \end^ = \begin \mathbf^ & \mathbf \\ -\mathbf^\mathbf^ & \mathbf^ \end.$

## By Neumann series

If a matrix A has the property that : $\lim_ \left(\mathbf I - \mathbf A\right)^n = 0$ then A is nonsingular and its inverse may be expressed by a
Neumann seriesA Neumann series is a series (mathematics), mathematical series of the form : \sum_^\infty T^k where ''T'' is an Operator (mathematics), operator and T^k := T^\circ its ''k'' times repeated application. This generalizes the geometric series. The s ...
: : $\mathbf A^ = \sum_^\infty \left(\mathbf I - \mathbf A\right)^n.$ Truncating the sum results in an "approximate" inverse which may be useful as a
preconditioner In mathematics, preconditioning is the application of a transformation, called the preconditioner, that conditions a given problem into a form that is more suitable for Numerical mathematics, numerical solving methods. Preconditioning is typical ...
. Note that a truncated series can be accelerated exponentially by noting that the Neumann series is a geometric sum. As such, it satisfies : $\sum_^ \left(\mathbf I - \mathbf A\right)^n = \prod_^\left\left(\mathbf I + \left(\mathbf I - \mathbf A\right)^\right\right)$. Therefore, only $2L - 2$ matrix multiplications are needed to compute $2^L$ terms of the sum. More generally, if A is "near" the invertible matrix X in the sense that : $\lim_ \left\left(\mathbf I - \mathbf X^ \mathbf A\right\right)^n = 0 \mathrm \lim_ \left\left(\mathbf I - \mathbf A \mathbf X^\right\right)^n = 0$ then A is nonsingular and its inverse is : $\mathbf A^ = \sum_^\infty \left\left(\mathbf X^ \left(\mathbf X - \mathbf A\right)\right\right)^n \mathbf X^~.$ If it is also the case that has
rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking A ranking is a relationship between a set of items such that, for any two items, the first is either "rank ...
1 then this simplifies to : $\mathbf A^ = \mathbf X^ - \frac~.$

If ''A'' is a matrix with
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
or
rational Rationality is the quality or state of being rational – that is, being based on or agreeable to reason Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογι ...
coefficients and we seek a solution in arbitrary-precision rationals, then a ''p''-adic approximation method converges to an exact solution in $O\left\left(n^4 \log^2 n\right\right)$, assuming standard $O\left\left(n^3\right\right)$ matrix multiplication is used. The method relies on solving ''n'' linear systems via Dixon's method of ''p''-adic approximation (each in $O\left(n^3 \log^2 n\right)$) and is available as such in software specialized in arbitrary-precision matrix operations, for example, in IML.

## Reciprocal basis vectors method

Given an $n \times n$ square matrix , $1 \leq i,j \leq n$, with $n$ rows interpreted as $n$ vectors $\mathbf_ = x^ \mathbf_$ (
Einstein summation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
assumed) where the $\mathbf_$ are a standard
orthonormal basis In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
of
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
$\mathbb^$ ($\mathbf_ = \mathbf^, \mathbf_ \cdot \mathbf^ = \delta_i^j$), then using
Clifford algebra In mathematics, a Clifford algebra is an algebra over a field, algebra generated by a vector space with a quadratic form, and is a Unital algebra, unital associative algebra. As algebra over a field, ''K''-algebras, they generalize the real nu ...
(or
Geometric Algebra In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
) we compute the reciprocal (sometimes called
dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality ** . . . see more cases in :Duality theories * Dual ...
) column vectors $\mathbf^ = x_ \mathbf^ = \left(-1\right)^ \left(\mathbf_ \wedge\cdots\wedge \left(\right)_ \wedge\cdots\wedge\mathbf_\right) \cdot \left(\mathbf_ \wedge\ \mathbf_ \wedge\cdots\wedge\mathbf_\right)^$ as the columns of the inverse matrix

# 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 : $\frac = - \mathbf^ \frac \mathbf^.$ To derive the above expression for the derivative of the inverse of A, one can differentiate the definition of the matrix inverse $\mathbf^\mathbf=\mathbf$ and then solve for the inverse of A: : $\frac = \frac\mathbf + \mathbf^\frac = \frac = \mathbf.$ Subtracting $\mathbf^\frac$ from both sides of the above and multiplying on the right by $\mathbf^$ gives the correct expression for the derivative of the inverse: : $\frac = - \mathbf^ \frac \mathbf^.$ Similarly, if $\varepsilon$ is a small number then : $\left\left(\mathbf + \varepsilon\mathbf\right\right)^ = \mathbf^ - \varepsilon \mathbf^ \mathbf \mathbf^ + \mathcal\left(\varepsilon^2\right)\,.$ More generally, if : $\frac = \sum_i g_i \left(\mathbf\right) \frach_i \left(\mathbf\right),$ then, : $f \left(\mathbf + \varepsilon\mathbf\right) = f \left(\mathbf\right) + \varepsilon\sum_i g_i \left(\mathbf\right) \mathbf h_i \left(\mathbf\right) + \mathcal\left\left(\varepsilon^2\right\right).$ Given a positive integer $n$, : $\begin \frac &= \sum_^n \mathbf^\frac\mathbf^,\\ \frac &= -\sum_^n \mathbf^\frac\mathbf^. \end$ Therefore, : $\begin \left(\mathbf + \varepsilon \mathbf\right)^ &= \mathbf^ + \varepsilon \sum_^n \mathbf^\mathbf\mathbf^ + \mathcal\left\left(\varepsilon^2\right\right),\\ \left(\mathbf + \varepsilon \mathbf\right)^ &= \mathbf^ - \varepsilon \sum_^n \mathbf^\mathbf\mathbf^ + \mathcal\left\left(\varepsilon^2\right\right). \end$

# Generalized inverse

Some of the properties of inverse matrices are shared by generalized inverses (for example, the
Moore–Penrose inverseIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
), 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.

## 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 an ''N'' × ''M'' 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.