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

where In 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−1. It follows from the theory of matrices that if

for square matrices A and B, then also

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. 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 commutative ring. 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 if you pick a random square matrix, it will almost surely not be singular. Matrix inversion is the process of finding the matrix B that satisfies the prior equation for a given invertible matrix A.

Properties of invertible matrices

Let A be a square n by n matrix over a field K (for example the field R of real numbers). Then the following statements are equivalent: A is invertible.

A is row-equivalent to the n-by-n identity matrix In.

A is column-equivalent to the n-by-n identity matrix In.

A has n pivot positions.

det A ≠ 0.

rank A = n.

The equation Ax = 0 has only the trivial solution x = 0 (i.e., Null A = {0}) The equation Ax = b has exactly one solution for each b in Kn. The columns of A are linearly independent.

The columns of A span Kn (i.e. Col A = Kn).

The columns of A form a basis of Kn.

The linear transformation mapping x to Ax is a bijection from Kn to Kn. There is an n by n matrix B such that AB = In.

The transpose AT is an invertible matrix.

The number 0 is not an eigenvalue of A.

The matrix A can expressed as a finite product of elementary matrices. In general, a square matrix over a commutative ring is invertible if and only if its determinant is a unit in that ring. *

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A matrix that is its own inverse, i.e. and , is called an involution.

Proof for matrix product rule

If A1, A2, ..., An 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 Ais from first principles, that is, that we wish to determine B such that

where B is the inverse matrix of the product. To remove A1 from the product, we can then write

which would reduce the equation to

Likewise, then, from

which simplifies to

If one repeats the process up to An, the equation becomes

but B is the inverse matrix, i.e. so the property is established

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 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). Analytic solution

Writing the transpose of the matrix of cofactors,...