Invertible matrix - wikipedia Invertible matrix This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. Please help to improve this article by introducing more precise citations. ( 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 A B = B A = I n ( displaystyle mathbf ( AB ) = mathbf ( BA ) = mathbf ( I ) _ ( n ) ) where I 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. 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 m ≠ n ) do not have an inverse. However, in some cases such a matrix may have a left inverse or right inverse. If A is m - by - n and the 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. 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 or complex 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 n × n invertible matrices together with the operation of matrix multiplication form a group, the general linear group of degree n. Contents 1 Properties 1.1 The invertible matrix theorem 1.2 Other properties 1.3 In relation to its adjugate 1.4 In relation to the identity matrix 1.5 Density 2 Examples 3 Methods of matrix inversion 3.1 Gaussian elimination 3.2 Newton 's method 3.3 Cayley -- Hamilton method 3.4 Eigendecomposition 3.5 Cholesky decomposition 3.6 Analytic solution 3.6. 1 Inversion of 2 × 2 matrices 3.6. 2 Inversion of 3 × 3 matrices 3.6. 3 Inversion of 4 × 4 matrices 3.7 Blockwise inversion 3.8 By Neumann series 3.9 P - adic approximation 4 Derivative of the matrix inverse 5 Generalized inverse 6 Applications 6.1 Regression / least squares 6.2 Matrix inverses in real - time simulations 6.3 Matrix inverses in MIMO wireless communication 7 See also 8 Notes 9 References 10 Further reading 11 External links Properties The invertible matrix theorem Let A be a square n by n matrix over a field K ( for example the field R of real numbers ). The following statements are equivalent, i.e., for any given matrix they are either all true or all false : A is invertible, i.e. A has an inverse, is nonsingular, or is nondegenerate. A is row - equivalent to the n - by - n identity matrix I. A is column - equivalent to the n - by - n identity matrix I. A has n pivot positions. det A ≠ 0. In general, a square matrix over a commutative ring is invertible if and only if its determinant is a unit in that ring. A has full rank ; that is, rank A = n. The equation Ax = 0 has only the trivial solution x = 0. Null A = ( 0 ). The equation Ax = b has exactly one solution for each b in K. The columns of A are linearly independent. The columns of A span K. Col A = K. The columns of A form a basis of K. The linear transformation mapping x to Ax is a bijection from K to K. There is an n - by - n matrix B such that AB = I = BA. The transpose A is an invertible matrix ( hence rows of A are linearly independent, span K, and form a basis of K ). The number 0 is not an eigenvalue of A. The matrix A can be expressed as a finite product of elementary matrices. The matrix A has a left inverse ( i.e. there exists a B such that BA = I ) or a right inverse ( i.e. there exists a C such that AC = I ), in which case both left and right inverses exist and B = C = A. Other properties Furthermore, the following properties hold for an invertible matrix A : ( A ) = A ; ( kA ) = k A for nonzero scalar k ; ( Ax ) = x A where denotes the Moore -- Penrose inverse and x is a vector ; ( A ) = ( A ) ; For any invertible n - by - n matrices A and B, ( AB ) = B A. More generally, if A,..., A are invertible n - by - n matrices, then ( A A ⋅ ⋅ ⋅ A A ) = A ⋯ A ; det A = ( det A ). A matrix that is its own inverse, i.e. such that A = A and A = I, is called an involutory matrix. In relation to its adjugate The adjugate of a matrix A ( displaystyle A ) can be used to find the inverse of A ( displaystyle A ) as follows : If A ( displaystyle A ) is an n × n ( displaystyle n \times n ) invertible matrix, then A − 1 = 1 det ( A ) adj ⁡ ( A ) ( displaystyle A ^ ( - 1 ) = ( \frac ( 1 ) ( det ( A ) ) ) operatorname ( adj ) ( A ) ). In relation to the identity matrix It follows from the associativity of matrix multiplication that if A B = I ( displaystyle mathbf ( AB ) = mathbf ( I ) ) for finite square matrices A and B, then also B A = I ( displaystyle mathbf ( BA ) = mathbf ( I ) ) Density Over the field of real numbers, the set of singular n - by - n matrices, considered as a subset of R, is a null set, i.e., has Lebesgue measure zero. This is true because singular matrices are the roots of the polynomial function in the entries of the matrix given by the determinant. 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 open set in the topological space 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 - conditioned. Examples Consider the following 2 - by - 2 matrix : A = ( − 1 3 2 1 − 1 ). ( displaystyle mathbf ( A ) = ( \begin ( pmatrix ) - 1& ( \tfrac ( 3 ) ( 2 ) ) 1& - 1 end ( pmatrix ) ). ) The matrix A ( displaystyle mathbf ( A ) ) is invertible. To check this, one can compute that det A = − 1 / 2 ( displaystyle det mathbf ( A ) = - 1 / 2 ), which is non-zero. As an example of a non-invertible, or singular, matrix, consider the matrix B = ( − 1 3 2 2 3 − 1 ). ( displaystyle mathbf ( B ) = ( \begin ( pmatrix ) - 1& ( \tfrac ( 3 ) ( 2 ) ) ( \tfrac ( 2 ) ( 3 ) ) & - 1 end ( pmatrix ) ). ) The determinant of B ( displaystyle mathbf ( B ) ) is 0, which is a necessary and sufficient condition for a matrix to be non-invertible. Methods of matrix inversion Gaussian elimination 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 algorithm may be convenient, if it is convenient to find a suitable starting seed : X k + 1 = 2 X k − X k A X k. ( displaystyle X_ ( k + 1 ) = 2X_ ( k ) - X_ ( k ) AX_ ( k ). ) Victor Pan and John Reif have done work that includes ways of generating a starting seed. Byte magazine 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, e.g. 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 allows the inverse of A to be expressed in terms of det ( A ), traces and powers of A A − 1 = 1 det ( A ) ∑ s = 0 n − 1 A s ∑ k 1, k 2,..., k n − 1 ∏ l = 1 n − 1 ( − 1 ) k l + 1 l k l k l ! t r ( A l ) k l, ( displaystyle 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 ) ! ) ) mathrm ( tr ) ( mathbf ( A ) ^ ( l ) ) ^ ( k_ ( l ) ), ) where n is dimension of A, and tr ( A ) is the trace of matrix A given by the sum of the main diagonal. The sum is taken over s and the sets of all k ≥ 0 satisfying the linear Diophantine equation s + ∑ l = 1 n − 1 l k l = n − 1. ( displaystyle s+ sum _ ( l = 1 ) ^ ( n - 1 ) lk_ ( l ) = n - 1. ) The formula can be rewritten in terms of complete Bell polynomials of arguments t = - ( l - 1 ) ! tr ( A ) as A − 1 = 1 det ( A ) ∑ s = 1 n A s − 1 ( − 1 ) n − 1 ( n − s ) ! B n − s ( t 1, t 2,..., t n − s ). ( displaystyle 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 ) ). ) 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 A − 1 = Q Λ − 1 Q − 1 ( displaystyle mathbf ( A ) ^ ( - 1 ) = mathbf ( Q ) mathbf ( Lambda ) ^ ( - 1 ) mathbf ( Q ) ^ ( - 1 ) ) where Q is the square ( N × N ) matrix whose i column is the eigenvector q i ( displaystyle q_ ( i ) ) of A and Λ is the diagonal matrix whose diagonal elements are the corresponding eigenvalues, i.e., Λ i i = λ i ( displaystyle Lambda _ ( ii ) = lambda _ ( i ) ). Furthermore, because Λ is a diagonal matrix, its inverse is easy to calculate : ( Λ − 1 ) i i = 1 λ i ( displaystyle left ( Lambda ^ ( - 1 ) \right ) _ ( ii ) = ( \frac ( 1 ) ( lambda _ ( i ) ) ) ) Cholesky decomposition Main article : Cholesky decomposition If matrix A is positive definite, then its inverse can be obtained as A − 1 = ( L ∗ ) − 1 L − 1, ( displaystyle mathbf ( A ) ^ ( - 1 ) = ( mathbf ( L ) ^ ( * ) ) ^ ( - 1 ) mathbf ( L ) ^ ( - 1 ), ) 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 : A − 1 = 1 A C T = 1 A ( C 11 C 21 ⋯ C n 1 C 12 C 22 ⋯ C n 2 ⋮ ⋮ ⋱ ⋮ C 1 n C 2 n ⋯ C n n ) ( displaystyle 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 ) ) ) so that ( A − 1 ) i j = 1 A ( C T ) i j = 1 A ( C j i ) ( displaystyle 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 ) ) where A is the determinant of A, C is the matrix of cofactors, and C represents the matrix transpose. Inversion of 2 × 2 matrices The cofactor equation listed above yields the following result for 2 × 2 matrices. Inversion of these matrices can be done as follows : A − 1 = ( a b c d ) − 1 = 1 det A ( d − b − c a ) = 1 a d − b c ( d − b − c a ). ( displaystyle 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 ) ). ) This is possible because 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 A − 1 = 1 det A ( ( tr ⁡ A ) I − A ). ( displaystyle mathbf ( A ) ^ ( - 1 ) = ( \frac ( 1 ) ( det mathbf ( A ) ) ) left ( left ( operatorname ( tr ) mathbf ( A ) \right ) mathbf ( I ) - mathbf ( A ) \right ). ) Inversion of 3 × 3 matrices A computationally efficient 3 × 3 matrix inversion is given by A − 1 = ( a b c d e f g h i ) − 1 = 1 det ( A ) ( A B C D E F G H I ) T = 1 det ( A ) ( A D G B E H C F I ) ( displaystyle 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 ) ) ) ( 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 A = ( e i − f h ) D = − ( b i − c h ) G = ( b f − c e ) B = − ( d i − f g ) E = ( a i − c g ) H = − ( a f − c d ) C = ( d h − e g ) F = − ( a h − b g ) I = ( a e − b d ) ( displaystyle ( \begin ( matrix ) A = ( ei - fh ) &D = - ( bi-ch ) &G = ( bf - ce ) B = - ( di - fg ) &E = ( ai - cg ) &H = - ( af - cd ) C = ( dh - eg ) &F = - ( ah - bg ) &I = ( ae - bd ) end ( matrix ) ) ) The determinant of A can be computed by applying the rule of Sarrus as follows : det ( A ) = a A + b B + c C. ( displaystyle det ( mathbf ( A ) ) = aA + bB + cC. ) The Cayley -- Hamilton decomposition gives A − 1 = 1 det ( A ) ( 1 2 ( ( tr ⁡ A ) 2 − tr ⁡ A 2 ) I − A tr ⁡ A + A 2 ). ( displaystyle 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 ). ) The general 3 × 3 inverse can be expressed concisely in terms of the cross product and triple product. If a matrix A = ( x 0, x 1, x 2 ) ( displaystyle mathbf ( A ) = left ( mathbf ( x ) _ ( 0 ), ; mathbf ( x ) _ ( 1 ), ; mathbf ( x ) _ ( 2 ) \right ) ) ( consisting of three column vectors, x 0 ( displaystyle mathbf ( x ) _ ( 0 ) ), x 1 ( displaystyle mathbf ( x ) _ ( 1 ) ), and x 2 ( displaystyle mathbf ( x ) _ ( 2 ) ) ) is invertible, its inverse is given by A − 1 = 1 det ( A ) ( ( x 1 × x 2 ) T ( x 2 × x 0 ) T ( x 0 × x 1 ) T ). ( displaystyle 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 ) ). ) Note that det ( A ) ( displaystyle det ( A ) ) is equal to the triple product of x 0 ( displaystyle mathbf ( x_ ( 0 ) ) ), x 1 ( displaystyle mathbf ( x_ ( 1 ) ) ), and x 2 ( displaystyle mathbf ( x_ ( 2 ) ) ) -- the volume of the parallelepiped formed by the rows or columns : det ( A ) = x 0 ⋅ ( x 1 × x 2 ). ( displaystyle det ( mathbf ( A ) ) = mathbf ( x ) _ ( 0 ) cdot ( mathbf ( x ) _ ( 1 ) \times mathbf ( x ) _ ( 2 ) ). ) 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 A − 1 ( displaystyle mathbf ( A ) ^ ( - 1 ) ) is orthogonal to the non-corresponding two columns of A ( displaystyle mathbf ( A ) ) ( causing the off - diagonal terms of I = A − 1 A ( displaystyle mathbf ( I ) = mathbf ( A ) ^ ( - 1 ) mathbf ( A ) ) be zero ). Dividing by det ( A ) = x 0 ⋅ ( x 1 × x 2 ) ( displaystyle det ( mathbf ( A ) ) = mathbf ( x ) _ ( 0 ) cdot ( mathbf ( x ) _ ( 1 ) \times mathbf ( x ) _ ( 2 ) ) ) causes the diagonal elements of I = A − 1 A ( displaystyle mathbf ( I ) = mathbf ( A ) ^ ( - 1 ) mathbf ( A ) ) to be unity. For example, the first diagonal is : 1 = 1 x 0 ⋅ ( x 1 × x 2 ) x 0 ⋅ ( x 1 × x 2 ). ( displaystyle 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 ) ). ) Inversion of 4 × 4 matrices With increasing dimension, expressions for the inverse of A get complicated. For n = 4, the Cayley -- Hamilton method leads to an expression that is still tractable : A − 1 = 1 det ( A ) ( 1 6 ( ( tr ⁡ A ) 3 − 3 tr ⁡ A tr ⁡ A 2 + 2 tr ⁡ A 3 ) I − 1 2 A ( ( tr ⁡ A ) 2 − tr ⁡ A 2 ) + A 2 tr ⁡ A − A 3 ). ( displaystyle 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 ). ) Blockwise inversion Matrices can also be inverted blockwise by using the following analytic inversion formula : ( A B C D ) − 1 = ( A − 1 + A − 1 B ( D − C A − 1 B ) − 1 C A − 1 − A − 1 B ( D − C A − 1 B ) − 1 − ( D − C A − 1 B ) − 1 C A − 1 ( D − C A − 1 B ) − 1 ), ( displaystyle ( \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 ) ), ) ( 1 ) 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 D − CA B must be nonsingular. ) This strategy is particularly advantageous if A is diagonal and D − CA 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 ), 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 ( 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 A − BD C are nonsingular, the result is ( A B C D ) − 1 = ( ( A − B D − 1 C ) − 1 − ( A − B D − 1 C ) − 1 B D − 1 − D − 1 C ( A − B D − 1 C ) − 1 D − 1 + D − 1 C ( A − B D − 1 C ) − 1 B D − 1 ). ( displaystyle ( \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 ) ). ) ( 2 ) Equating Equations ( 1 ) and ( 2 ) leads to ( A − B D − 1 C ) − 1 = A − 1 + A − 1 B ( D − C A − 1 B ) − 1 C A − 1 ( displaystyle ( 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 ) , ) ( 3 ) ( A − B D − 1 C ) − 1 B D − 1 = A − 1 B ( D − C A − 1 B ) − 1 ( displaystyle ( mathbf ( A ) - mathbf ( BD ) ^ ( - 1 ) mathbf ( C ) ) ^ ( - 1 ) mathbf ( BD ) ^ ( - 1 ) = mathbf ( A ) ^ ( - 1 ) mathbf ( B ) ( mathbf ( D ) - mathbf ( CA ) ^ ( - 1 ) mathbf ( B ) ) ^ ( - 1 ) , ) D − 1 C ( A − B D − 1 C ) − 1 = ( D − C A − 1 B ) − 1 C A − 1 ( displaystyle mathbf ( D ) ^ ( - 1 ) mathbf ( C ) ( mathbf ( A ) - mathbf ( BD ) ^ ( - 1 ) mathbf ( C ) ) ^ ( - 1 ) = ( mathbf ( D ) - mathbf ( CA ) ^ ( - 1 ) mathbf ( B ) ) ^ ( - 1 ) mathbf ( CA ) ^ ( - 1 ) , ) D − 1 + D − 1 C ( A − B D − 1 C ) − 1 B D − 1 = ( D − C A − 1 B ) − 1 ( displaystyle 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 ) , ) where Equation ( 3 ) is the Woodbury matrix identity, which is equivalent to the binomial inverse theorem. Since a blockwise inversion of an n × 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. There exist matrix multiplication algorithms with a complexity of O ( n ) operations, while the best proven lower bound is Ω ( n log n ). By Neumann series If a matrix A has the property that lim n → ∞ ( I − A ) n = 0 ( displaystyle lim _ ( n \to infty ) ( mathbf ( I ) - mathbf ( A ) ) ^ ( n ) = 0 ) then A is nonsingular and its inverse may be expressed by a Neumann series : A − 1 = ∑ n = 0 ∞ ( I − A ) n. ( displaystyle mathbf ( A ) ^ ( - 1 ) = sum _ ( n = 0 ) ^ ( infty ) ( mathbf ( I ) - mathbf ( A ) ) ^ ( n ). ) 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 ∑ n = 0 2 L − 1 ( I − A ) n = ∏ l = 0 L − 1 ( I + ( I − A ) 2 l ) ( displaystyle sum _ ( n = 0 ) ^ ( 2 ^ ( L ) - 1 ) ( mathbf ( I ) - mathbf ( A ) ) ^ ( n ) = prod _ ( l = 0 ) ^ ( L - 1 ) ( mathbf ( I ) + ( mathbf ( I ) - mathbf ( A ) ) ^ ( 2 ^ ( l ) ) ) ). Therefore, only 2 L − 2 ( displaystyle 2L - 2 ) matrix multiplications are needed to compute 2 L ( displaystyle 2 ^ ( L ) ) terms of the sum. More generally, if A is `` near '' the invertible matrix X in the sense that lim n → ∞ ( I − X − 1 A ) n = 0 o r lim n → ∞ ( I − A X − 1 ) n = 0 ( displaystyle 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 ) then A is nonsingular and its inverse is A − 1 = ∑ n = 0 ∞ ( X − 1 ( X − A ) ) n X − 1. ( displaystyle mathbf ( A ) ^ ( - 1 ) = sum _ ( n = 0 ) ^ ( infty ) left ( mathbf ( X ) ^ ( - 1 ) ( mathbf ( X ) - mathbf ( A ) ) \right ) ^ ( n ) mathbf ( X ) ^ ( - 1 ) ~. ) If it is also the case that A − X has rank 1 then this simplifies to A − 1 = X − 1 − X − 1 ( A − X ) X − 1 1 + tr ⁡ ( X − 1 ( A − X ) ). ( displaystyle 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 ) ) ) ) ) ~. ) P - adic approximation This section needs expansion. You can help by adding to it. ( February 2015 ) If A is a matrix with integer or rational coefficients and we seek a solution in arbitrary - precision rationals, then a p - adic approximation method converges to an exact solution in O ( n 4 log 2 ⁡ n ) ( displaystyle O ( n ^ ( 4 ) log ^ ( 2 ) n ) ), assuming standard O ( n 3 ) ( displaystyle O ( n ^ ( 3 ) ) ) matrix multiplication is used. The method relies on solving n linear systems via Dixon 's method of p - adic approximation ( each in O ( n 3 log 2 ⁡ n ) ( displaystyle O ( n ^ ( 3 ) log ^ ( 2 ) n ) ) ) and is available as such in software specialized in arbitrary - precision matrix operations, e.g. in IML. 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 d A − 1 d t = − A − 1 d A d t A − 1. ( displaystyle ( \frac ( mathrm ( d ) mathbf ( A ) ^ ( - 1 ) ) ( mathrm ( d ) t ) ) = - mathbf ( A ) ^ ( - 1 ) ( \frac ( mathrm ( d ) mathbf ( A ) ) ( mathrm ( d ) t ) ) mathbf ( A ) ^ ( - 1 ). ) To derive the above expression for the derivative of the inverse of A, one can differentiate the definition of the matrix inverse A − 1 A = I ( displaystyle mathbf ( A ) ^ ( - 1 ) mathbf ( A ) = mathbf ( I ) ) and then solve for the inverse of A : d A − 1 A d t = d A − 1 d t A + A − 1 d A d t = d I d t = 0. ( displaystyle ( \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 ). ) Subtracting A − 1 d A d t ( displaystyle mathbf ( A ) ^ ( - 1 ) ( \frac ( mathrm ( d ) mathbf ( A ) ) ( mathrm ( d ) t ) ) ) from both sides of the above and multiplying on the right by A − 1 ( displaystyle mathbf ( A ) ^ ( - 1 ) ) gives the correct expression for the derivative of the inverse : d A − 1 d t = − A − 1 d A d t A − 1. ( displaystyle ( \frac ( mathrm ( d ) mathbf ( A ) ^ ( - 1 ) ) ( mathrm ( d ) t ) ) = - mathbf ( A ) ^ ( - 1 ) ( \frac ( mathrm ( d ) mathbf ( A ) ) ( mathrm ( d ) t ) ) mathbf ( A ) ^ ( - 1 ). ) Similarly, if ε ( displaystyle epsilon ) is a small number then ( A + ε X ) − 1 = A − 1 − ε A − 1 X A − 1 + O ( ε 2 ). ( displaystyle left ( mathbf ( A ) + epsilon mathbf ( X ) \right ) ^ ( - 1 ) = mathbf ( A ) ^ ( - 1 ) - epsilon mathbf ( A ) ^ ( - 1 ) mathbf ( X ) mathbf ( A ) ^ ( - 1 ) + ( mathcal ( O ) ) ( epsilon ^ ( 2 ) ) ,. ) More generally, if d f ( A ) d t = ∑ i g i ( A ) d A d t h i ( A ), ( displaystyle ( \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 ) ), ) then, f ( A + ε X ) = f ( A ) + ε ∑ i g i ( A ) X h i ( A ) + O ( ε 2 ). ( displaystyle 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 ) ). ) Given a positive integer n ( displaystyle n ), d A n d t = ∑ i = 1 n A i − 1 d A d t A n − i, d A − n d t = − ∑ i = 1 n A − i d A d t A − ( n + 1 − i ). ( displaystyle ( \begin ( aligned ) ( \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 ( aligned ) ) ) Therefore, ( A + ε X ) n = A n + ε ∑ i = 1 n A i − 1 X A n − i + O ( ε 2 ), ( A + ε X ) − n = A − n − ε ∑ i = 1 n A − i X A − ( n + 1 − i ) + O ( ε 2 ). ( displaystyle ( \begin ( aligned ) ( 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 ( aligned ) ) ) Generalized inverse Some of the properties of inverse matrices are shared by generalized inverses ( e.g., 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 unavoidable to estimate their precision, found in the diagonal of the posterior covariance matrix of the vector of unknowns. 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 Notes Jump up ^ Horn, Roger A. ; Johnson, Charles R. ( 1985 ). Matrix Analysis. Cambridge University Press. p. 14. ISBN 978 - 0 - 521 - 38632 - 6... Jump up ^ Pan, Victor ; Reif, John ( 1985 ), Efficient Parallel Solution of Linear Systems, Proceedings of the 17th Annual ACM Symposium on Theory of Computing, Providence : ACM Jump up ^ Pan, Victor ; Reif, John ( 1985 ), Harvard University Center for Research in Computing Technology Report TR - 02 - 85, Cambridge, MA : Aiken Computation Laboratory Jump up ^ `` The Inversion of Large Matrices ''. Byte magazine. 11 ( 04 ) : 181 -- 190. April 1986. Jump up ^ A proof can be found in the Appendix B of Kondratyuk, L.A. ; Krivoruchenko, M.I. ( 1992 ). `` Superconducting quark matter in SU ( 2 ) color group ''. Zeitschrift für Physik A. 344 : 99 -- 115. doi : 10.1007 / BF01291027. Jump up ^ Strang, Gilbert ( 2003 ). Introduction to linear algebra ( 3rd ed. ). SIAM. p. 71. ISBN 0 - 9614088 - 9 - 8., Chapter 2, page 71 Jump up ^ Bernstein, Dennis ( 2005 ). Matrix Mathematics. Princeton University Press. p. 44. ISBN 0 - 691 - 11802 - 7. Jump up ^ Bernstein, Dennis ( 2005 ). Matrix Mathematics. Princeton University Press. p. 45. ISBN 0 - 691 - 11802 - 7. Jump up ^ T.H. Cormen, C.E. Leiserson, R.L. Rivest, C. Stein, Introduction to Algorithms, 3rd ed., MIT Press, Cambridge, MA, 2009, § 28.2. Jump up ^ 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. Jump up ^ Stewart, Gilbert ( 1998 ). Matrix Algorithms : Basic decompositions. SIAM. p. 55. ISBN 0 - 89871 - 414 - 1. Jump up ^ `` A p - adic algorithm for computing the inverse of integer matrices ''. Journal of Computational and Applied Mathematics. 225 : 320 -- 322. doi : 10.1016 / j. cam. 2008.07. 044. Jump up ^ `` IML - Integer Matrix Library ''. cs.uwaterloo.ca. Retrieved 14 April 2018. References Cormen, Thomas H. ; Leiserson, Charles E. ; Rivest, Ronald L. ; Stein, Clifford ( 2001 ) ( 1990 ). `` 28.4 : Inverting matrices ''. Introduction to Algorithms ( 2nd ed. ). MIT Press and McGraw - Hill. pp. 755 -- 760. ISBN 0 - 262 - 03293 - 7. Further reading Hazewinkel, Michiel, ed. ( 2001 ) ( 1994 ), `` Inversion of a matrix '', Encyclopedia of Mathematics, Springer Science + Business Media B.V. / Kluwer Academic Publishers, ISBN 978 - 1 - 55608 - 010 - 4 Matrix Mathematics : Theory, Facts, and Formulas at Google books The Matrix Cookbook External links This article 's use of external links may not follow Wikipedia 's policies or guidelines. Please improve this article by removing excessive or inappropriate external links, and converting useful links where appropriate into footnote references. ( June 2015 ) Lecture on Inverse Matrices by Khan Academy Linear Algebra Lecture on Inverse Matrices by MIT LAPACK -- a collection of FORTRAN subroutines for solving dense linear algebra problems ALGLIB -- includes a partial port of the LAPACK to C++, C#, Delphi, etc. Online Inverse Matrix Calculator using AJAX Symbolic Inverse of Matrix Calculator with steps shown Moore - Penrose Pseudo-Inverse Matrix Inverse of a Matrix Notes Calculator for Singular or Non-Square Matrix Inverse Inverse calculator online `` Derivative of inverse matrix ''. 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Question: when does the inverse of a square matrix not exist
Answer: singular or degenerate