Invertible matrix
In linear algebra, an nbyn square matrix A is called invertible (also nonsingular or nondegenerate), if there exists an nbyn square matrix B such that
where I_{n} denotes the nbyn 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 (multiplicative) inverse of A, denoted by A^{−1}.[1][2] Matrix inversion is the process of finding the matrix B that satisfies the prior equation for a given invertible matrix 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 zero.[3] 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. Nonsquare matrices (mbyn 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 mbyn and the rank of A is equal to n (n ≤ m), then A has a left inverse, an nbym matrix B such that BA = I_{n}. If A has rank m (m ≤ n), then it has a right inverse, an nbym matrix B such that AB = I_{m}.
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 leftinverse or rightinverse 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 (and entries from ring R) form a group, the general linear group of degree n, denoted GL_{n}(R).[1]
Properties
The invertible matrix theorem
Let A be a square n by n matrix over a field 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):[4]
 A is invertible, that is, A has an inverse, is nonsingular, or is nondegenerate.
 A is rowequivalent to the nbyn identity matrix I_{n}.
 A is columnequivalent to the nbyn identity matrix I_{n}.
 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.
 The kernel of A is trivial, that is, it contains only the null vector as an element, ker(A) = {0}.
 The equation Ax = b has exactly one solution for each b in K^{n}.
 The columns of A are linearly independent.
 The columns of A span K^{n}.
 Col A = K^{n}.
 The columns of A form a basis of K^{n}.
 The linear transformation mapping x to Ax is a bijection from K^{n} to K^{n}.
 There is an nbyn matrix B such that AB = I_{n} = BA.
 The transpose A^{T} is an invertible matrix (hence rows of A are linearly independent, span K^{n}, and form a basis of K^{n}).
 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 (that is, there exists a B such that BA = I) or a right inverse (that is, there exists a C such that AC = I), in which case both left and right inverses exist and B = C = A^{−1}.
Other properties
Furthermore, the following properties hold for an invertible matrix A:
 (A^{−1})^{−1} = A;
 (kA)^{−1} = k^{−1}A^{−1} for nonzero scalar k;
 (Ax)^{+} = x^{+}A^{−1} if A has orthonormal columns, where ^{+} denotes the Moore–Penrose inverse and x is a vector;
 (A^{T})^{−1} = (A^{−1})^{T};
 For any invertible nbyn matrices A and B, (AB)^{−1} = B^{−1}A^{−1}. More generally, if A_{1}, ..., A_{k} are invertible nbyn matrices, then (A_{1}A_{2}⋯A_{k−1}A_{k})^{−1} = A^{−1}
_{k}A^{−1}
_{k−1}⋯A^{−1}
_{2}A^{−1}
_{1};  det A^{−1} = (det A)^{−1}.
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 the rows of V are denoted as and the columns of U as for . Then clearly, the Euclidean inner product of any two . 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. In which case, one can apply the iterative Gram–Schmidt process 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 A = A^{−1} and A^{2} = I), is called an involutory matrix.
In relation to its adjugate
The adjugate of a matrix can be used to find the inverse of as follows:
If is an invertible matrix, then
In relation to the identity matrix
It follows from the associativity of matrix multiplication that if
for finite square matrices A and B, then also
Density
Over the field of real numbers, the set of singular nbyn matrices, considered as a subset of R^{n×n}, is a null set, that is, has 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 nbyn matrices are invertible.
Furthermore, the nbyn invertible matrices are a dense open set in the topological space of all nbyn matrices. Equivalently, the set of singular matrices is closed and nowhere dense in the space of nbyn matrices.
In practice however, one may encounter noninvertible matrices. And in numerical calculations, matrices which are invertible, but close to a noninvertible matrix, can still be problematic; such matrices are said to be illconditioned.
Examples
Consider the following 2by2 matrix:
The matrix is invertible. To check this, one can compute that , which is nonzero.
As an example of a noninvertible, or singular, matrix, consider the matrix
The determinant of is 0, which is a necessary and sufficient condition for a matrix to be noninvertible.
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:
Victor Pan and John Reif have done work that includes ways of generating a starting seed.[6][7] Byte magazine summarised one of their approaches.[8]
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 allows the inverse of to be expressed in terms of , traces and powers of :[9]
where is dimension of , and is the trace of matrix given by the sum of the main diagonal. The sum is taken over and the sets of all satisfying the linear Diophantine equation
The formula can be rewritten in terms of complete Bell polynomials of arguments as
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
where is the square (N×N) matrix whose ith column is the eigenvector of , and is the diagonal matrix whose diagonal elements are the corresponding eigenvalues, that is, . If is symmetric, is guaranteed to be an orthogonal matrix, therefore . Furthermore, because is a diagonal matrix, its inverse is easy to calculate:
Cholesky decomposition
If matrix A is positive definite, then its inverse can be obtained as
where L is the lower triangular Cholesky decomposition of A, and L* denotes the conjugate transpose of L.
Analytic solution
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:
so that
where A is the determinant of A, C is the matrix of cofactors, and C^{T} 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:[10]
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
Inversion of 3 × 3 matrices
A computationally efficient 3 × 3 matrix inversion is given by
(where the scalar A is not to be confused with the matrix A).
If the determinant is nonzero, the matrix is invertible, with the elements of the intermediary matrix on the right side above given by
The determinant of A can be computed by applying the rule of Sarrus as follows:
The Cayley–Hamilton decomposition gives
The general 3 × 3 inverse can be expressed concisely in terms of the cross product and triple product. If a matrix (consisting of three column vectors, , , and ) is invertible, its inverse is given by
The determinant of A, , is equal to the triple product of , , and —the volume of the parallelepiped formed by the rows or columns:
The correctness of the formula can be checked by using cross and tripleproduct properties and by noting that for groups, left and right inverses always coincide. Intuitively, because of the cross products, each row of is orthogonal to the noncorresponding two columns of (causing the offdiagonal terms of be zero). Dividing by
causes the diagonal elements of to be unity. For example, the first diagonal is:
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:
Blockwise inversion
Matrices can also be inverted blockwise by using the following analytic inversion formula:

(1)
where A, B, C and D are matrix subblocks of arbitrary size. (A must be square, so that it can be inverted. Furthermore, A and D − CA^{−1}B must be nonsingular.[11]) This strategy is particularly advantageous if A is diagonal and D − CA^{−1}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 subblock in the lower right of the inverse matrix, and that the nullity of B equals the nullity of the subblock 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^{−1}C are nonsingular,[12] the result is

(2)
Equating Equations (1) and (2) leads to

(3)
where Equation (3) is the Woodbury matrix identity, 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

(2)
By the Weinstein–Aronszajn identity, one of the two matrices in the blockdiagonal matrix is invertible exactly when the other is.
Since a blockwise inversion of an n × n matrix requires inversion of two halfsized matrices and 6 multiplications between two halfsized 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.[13] Research into matrix multiplication complexity shows that there exist matrix multiplication algorithms with a complexity of O(n^{2.3727}) operations, while the best proven lower bound is Ω(n^{2} log n).[14]
This formula simplifies significantly when the upper right block matrix is the zero matrix. This formulation is useful when the matrices and 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
By Neumann series
If a matrix A has the property that
then A is nonsingular and its inverse may be expressed by a Neumann series:[15]
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
 .
Therefore, only matrix multiplications are needed to compute terms of the sum.
More generally, if A is "near" the invertible matrix X in the sense that
then A is nonsingular and its inverse is
If it is also the case that A − X has rank 1 then this simplifies to
padic approximation
If A is a matrix with integer or rational coefficients and we seek a solution in arbitraryprecision rationals, then a padic approximation method converges to an exact solution in , assuming standard matrix multiplication is used.[16] The method relies on solving n linear systems via Dixon's method of padic approximation (each in ) and is available as such in software specialized in arbitraryprecision matrix operations, for example, in IML.[17]
Reciprocal basis vectors method
Given an square matrix , , with rows interpreted as vectors (Einstein summation assumed) where the are a standard orthonormal basis of Euclidean space (), then using Clifford algebra (or Geometric Algebra) we compute the reciprocal (sometimes called dual) column vectors as the columns of the inverse matrix . Note that, the place "" indicates that "" is removed from that place in the above expression for . We then have , where is the Kronecker delta. We also have , as required. If the vectors are not linearly independent, then and the matrix is not invertible (has no inverse).
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[18]
To derive the above expression for the derivative of the inverse of A, one can differentiate the definition of the matrix inverse and then solve for the inverse of A:
Subtracting from both sides of the above and multiplying on the right by gives the correct expression for the derivative of the inverse:
Similarly, if is a small number then
More generally, if
then,
Given a positive integer ,
Therefore,
Generalized inverse
Some of the properties of inverse matrices are shared by generalized inverses (for example, the Moore–Penrose inverse), which can be defined for any mbyn 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.[19]
Matrix inverses in realtime simulations
Matrix inversion plays a significant role in computer graphics, particularly in 3D graphics rendering and 3D simulations. Examples include screentoworld ray casting, worldtosubspacetoworld object transformations, and physical simulations.
Matrix inverses in MIMO wireless communication
Matrix inversion also plays a significant role in the MIMO (MultipleInput, MultipleOutput) 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.
See also
 Binomial inverse theorem
 LU decomposition
 Matrix decomposition
 Matrix square root
 Minor (linear algebra)
 Partial inverse of a matrix
 Pseudoinverse
 Singular value decomposition
 Woodbury matrix identity
References
 "Comprehensive List of Algebra Symbols". Math Vault. 20200325. Retrieved 20200908.
 "Invertible Matrices". www.sosmath.com. Retrieved 20200908.
 Weisstein, Eric W. "Matrix Inverse". mathworld.wolfram.com. Retrieved 20200908.
 Weisstein, Eric W. "Invertible Matrix Theorem". mathworld.wolfram.com. Retrieved 20200908.
 Horn, Roger A.; Johnson, Charles R. (1985). Matrix Analysis. Cambridge University Press. p. 14. ISBN 9780521386326..
 Pan, Victor; Reif, John (1985), Efficient Parallel Solution of Linear Systems, Proceedings of the 17th Annual ACM Symposium on Theory of Computing, Providence: ACM
 Pan, Victor; Reif, John (1985), Harvard University Center for Research in Computing Technology Report TR0285, Cambridge, MA: Aiken Computation Laboratory
 "The Inversion of Large Matrices". Byte Magazine. 11 (4): 181–190. April 1986.
 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. S2CID 120467300.
 Strang, Gilbert (2003). Introduction to linear algebra (3rd ed.). SIAM. p. 71. ISBN 9780961408893., Chapter 2, page 71
 Bernstein, Dennis (2005). Matrix Mathematics. Princeton University Press. p. 44. ISBN 9780691118024.
 Bernstein, Dennis (2005). Matrix Mathematics. Princeton University Press. p. 45. ISBN 9780691118024.
 T. H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein, Introduction to Algorithms, 3rd ed., MIT Press, Cambridge, MA, 2009, §28.2.
 Ran Raz. On the complexity of matrix product. In Proceedings of the thirtyfourth annual ACM symposium on Theory of computing. ACM Press, 2002. doi:10.1145/509907.509932.
 Stewart, Gilbert (1998). Matrix Algorithms: Basic decompositions. SIAM. p. 55. ISBN 9780898714142.
 Haramoto, H.; Matsumoto, M. (2009). "A padic 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.
 "IML  Integer Matrix Library". cs.uwaterloo.ca. Retrieved 14 April 2018.
 Magnus, Jan R.; Neudecker, Heinz (1999). Matrix Differential Calculus : with Applications in Statistics and Econometrics (Revised ed.). New York: John Wiley & Sons. pp. 151–152. ISBN 047198633X.
 Lin, Lin; Lu, Jianfeng; Ying, Lexing; Car, Roberto; E, Weinan (2009). "Fast algorithm for extracting the diagonal of the inverse matrix with application to the electronic structure analysis of metallic systems". Communications in Mathematical Sciences. 7 (3): 755–777. doi:10.4310/CMS.2009.v7.n3.a12.
Further reading
 "Inversion of a matrix", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
 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 McGrawHill. pp. 755–760. ISBN 0262032937.
 Bernstein, Dennis S. (2009). Matrix Mathematics: Theory, Facts, and Formulas (2nd ed.). Princeton University Press. ISBN 9780691140391 – via Google Books.
 Petersen, Kaare Brandt; Pedersen, Michael Syskind (November 15, 2012). "The Matrix Cookbook" (PDF). pp. 17–23.
External links
 Sanderson, Grant (August 15, 2016). "Inverse Matrices, Column Space and Null Space". Essence of Linear Algebra – via YouTube.
 Strang, Gilbert. "Linear Algebra Lecture on Inverse Matrices". MIT OpenCourseWare.
 Symbolic Inverse of Matrix Calculator with steps shown
 MoorePenrose Inverse Matrix