Invertible Matrix Invertible Matrix Theorem Further Properties

where I_n 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. A square matrix that is not invertible is called singular. 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.

1 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:

In general, a square matrix over a commutative ringIn ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. This means that if a and b are any elements of the ring, and if the multiplication operation is written as then a ' is invertible if and only if its determinant is a unitRing theory In mathematics, a unit in a ring R is an element u such that there is v in R with uv vu 1. That is, u is an invertible element of the multiplicative monoid of R''. The units of R form a group U ''R under multiplication, the group of units of R in that ring.

2 Further properties and facts

To check whether a given matrix is invertible, and to compute the inverse in small examples, one typically uses the Gauss-Jordan elimination algorithm. Other methods are explained under matrix inversionMatrix 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 the n by n identity matrix. The Gauss-Jordan elimination is an algorithm that can be used to determ.

The inverse of an invertible matrix A is itself invertible, with

(A⁻¹)⁻¹ = A

The inverse of an invertible matrix A multiplied by a scalar k yields the product of the inverse of both the matrix and the scalar

(kA)⁻¹ = k⁻¹A⁻¹

The product of two invertible matrices A and B of the same size is again invertible, with the inverse given by

(AB)⁻¹ = B⁻¹A⁻¹

(note that the order of the factors is reversed.) As a consequence, the set of invertible n-by-n matrices forms a groupIn mathematics, a group is a set, together with a binary operation satisfying certain axioms, detailed below. The branch of mathematics which studies groups is called group theory. The historical origin of group theory goes back to the works of Evariste G, known as the general linear group Gl(n).

As a rule of thumb, "almost all" matrices are invertible. Over the field of real numbers, this can be made precise as follows: the set of singular n-by-n matrices, considered as a subset of R^n×n, is a null set, i.e. has Lebesgue measure zero. Intuitively, this means that if you pick a random square matrix over the reals, the probability that it be singular is zero. The reason for this is that singular matrices can be thought of as the roots of the polynomial function given by the determinant.

3 Generalizations

Some of the properties of inverse matrices are shared by pseudoinverses which can be defined for every matrix, even for non-square ones.

4 External links

Moore Penrose Pseudoinverse

Matrices

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