In mathematics and especially linear algebra, an n-by-n matrix A is called invertible or non-singular if there exists another n-by-n matrix B such that
- AB = BA = In,
Table of contents |
2 Further properties and facts 3 Generalizations |
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 and must all be true for A to be invertible:
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 inversion.
The inverse of an invertible matrix A is itself invertible, with
Invertible Matrix Theorem
Further properties and facts
The product of two invertible matrices A and B of the same size is again invertible, with the inverse given by
(note that the order of the factors is reversed.) As a consequence, the set of invertible n-by-n matrices forms a group, 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 Rn×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.
A square matrix with entries from some commutative ring is invertible if and only if its determinant is a unit in that ring.