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Matrix inversion is the following problem in linear algebra: given a square n-by-n matrix A, find a square n-by-n matrix B (if one exists) such that AB = BA = I_n, the n-by-n identity matrix. Wikipedia does not have an article with this exact name. ... In mathematics and especially linear algebra, an n-by-n matrix A is called invertible, non-singular or regular if there exists another n-by-n matrix B such that AB = BA = In, where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. ... For the square matrix section, see square matrix. ... In linear algebra, the identity matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere. ...

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 an upper and a lower triangular matrices which are easier to invert. For special purposes, it may be convenient to invert matrices by treating mn-by-mn matrices as m-by-m matrices of n-by-n matrices, and applying one or another formula recursively (other sized matrices can be padded out with dummy rows and columns). For other purposes, a variant of Newton's method may be convenient (particularly when dealing with families of related matrices, so inverses of earlier matrices can be used to seed generating inverses of later matrices). In mathematics, Gaussian elimination or Gauss-Jordan elimination, named after Carl Friedrich Gauss and Wilhelm Jordan, is an algorithm in linear algebra for determining the solutions of a system of linear equations, for determining the rank of a matrix, and for calculating the inverse of an invertible square matrix. ... Flowcharts are often used to represent algorithms. ... In mathematics and especially linear algebra, an n-by-n matrix A is called invertible, non-singular or regular if there exists another n-by-n matrix B such that AB = BA = In, where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ... In linear algebra, a LU decomposition, or LUP decomposition is a matrix decomposition of a matrix into a lower triangular matrix L, an upper-triangular matrix U, and a permutation matrix P. This decomposition is used in numerical analysis to solve systems of linear equations. ... In numerical analysis, Newtons method (or the Newton-Raphson method) is an efficient algorithm for finding approximations to the zeros (or roots) of a real-valued function. ...

Analytic solution

Writing another special matrix of cofactors, known as an adjugate matrix, can also be an efficient way to calculate the inverse of small matrices (since this method is essentially recursive, it becomes inefficient for large matrices). To determine the inverse, we calculate a matrix of cofactors: In linear algebra, a minor of a matrix is the determinant of a certain smaller matrix. ... In linear algebra, the adjugate or classical adjoint of a square matrix is a matrix which plays a role similar to the inverse of a matrix; it can however be defined for any square matrix without the need to perform any divisions. ...

where |A| is the determinant of A, C_ij is the matrix cofactor, and A^T represents the matrix transpose. In linear algebra, a determinant is a function depending on n that associates a scalar det(A) to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... In mathematics, and in particular linear algebra, the transpose of a matrix is another matrix, produced by turning rows into columns and vice versa. ...

In most practical applications, it is in fact not necessary to invert a matrix to solve a system of linear equations. This can instead be done using decomposition techniques like LU decomposition, which are much faster than inversion. Various fast algorithms for special classes of linear systems have also been developed. In mathematics and linear algebra, a system of linear equations is a set of linear equations such as 3x1 + 2x2 − x3 = 1 2x1 − 2x2 + 4x3 = −2 −x1 + ½x2 − x3 = 0. ... In linear algebra, a LU decomposition, or LUP decomposition is a matrix decomposition of a matrix into a lower triangular matrix L, an upper-triangular matrix U, and a permutation matrix P. This decomposition is used in numerical analysis to solve systems of linear equations. ...

Inversion of 2 x 2 matrices

The cofactor equation listed above yields the following result for 2 x 2 matrices. Inversion of these matrices can be done easily as follows:

Inversion of 3 x 3 matrices

The cofactor equation listed above yields the following result for 3 x 3 matrices. Inversion of these matrices can be done quite easily as follows:

| A | = a(ei − fh) − b(di − fg) + c(dh − eg)

Blockwise inversion

Matrices can also be inverted blockwisely by using the following analytic inversion formula:

where A, B, C and D are matrix sub-blocks of arbitrary size. 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 to be inverted. In the mathematical discipline of matrix theory, a block matrix or a partitioned matrix is a partition of a matrix into rectangular smaller matrices called blocks. ... In linear algebra and the theory of matrices, the Schur complement (named after Issai Schur) of a block of a matrix within the larger matrix is defined as follows. ...

This technique was invented by Volker Strassen, who also invented the Strassen algorithm for fast(er) matrix multiplication. Volker Strassen is a German mathematician. ... In the mathematical discipline of linear algebra, the Strassen algorithm, named after Volker Strassen, is an algorithm used for matrix multiplication. ... This article gives an overview of the various ways to multiply matrices. ...

Properties

If A₁, A₂, ..., A_n are nonsingular square matrices over a field, then

It becomes evident why this is the case if one attempts to find an inverse for the product of the A_is from first principles, that is, that we wish to determine B such that

where B is some matrix, in terms of the A_is. To remove A_n from the product, we can then write

where B' is some matrix, which would reduce the equation to

Likewise, then, from

we use the same technique, removing A_{n − 1} from the equation, yielding

where B' is some matrix, which, when simplified, gives

If one repeat the process up to A₁, the above property is established.

See also

• matrix decomposition
• matrix multiplication
• pseudoinverse (Moore-Penrose inverse)
• singular value decomposition