In linear algebra, an invertible matrix is a square matrix that has an inverse. In other words, if some other matrix is multiplied by the invertible matrix, the result can be multiplied by an inverse to undo the operation. An invertible matrix multiplied by its inverse yields the identity matrix. Invertible matrices are the same size as their inverse.

An -by- square matrix is called invertible (also nonsingular, nondegenerate or rarely regular) if there exists an -by- square matrix such that

(1)

where denotes the -by- identity matrix and the multiplication used is ordinary matrix multiplication[ If this is the case, then the matrix is uniquely determined by , and is called the multiplicative inverse of , denoted by . Matrix inversion is the process of finding the matrix which when multiplied by the original matrix gives the identity matrix[2].

Over a field, a square matrix that is not invertible is called singular or degenerate. A square matrix with entries in a field is singular if and only if its determinant is zero. Singular matrices are rare in the sense that if a square matrix's entries are randomly selected from any bounded 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. Non-square matrices, i.e. -by- matrices for which , do not have an inverse. However, in some cases such a matrix may have a left inverse or right inverse. If is -by- and the rank of is equal to , then has a left inverse, an -by- matrix such that . If has rank , then it has a right inverse, an -by- matrix such that .

While the most common case is that of matrices over the real or complex numbers, all of those definitions can be given for matrices over any algebraic structure equipped with addition and multiplication (i.e. rings). However, in the case of a 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 it being nonzero. For a noncommutative ring, the usual determinant is not defined. The conditions for existence of left-inverse or right-inverse are more complicated, since a notion of rank does not exist over rings.

The set of invertible matrices together with the operation of matrix multiplication and entries from ring form a group, the general linear group of degree , denoted .

This article is a derivative work of the creative commons share alike with attribution in [3].

Bibliography

[1] "Inversion of a matrix", Encyclopedia of Mathematics, EMS Press, 2001 [1994]

[2] 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 McGraw-Hill. pp. 755-760. ISBN 0-262-03293-7.

[3] Wikipedia contributors, "Invertible matrix," Wikipedia, The Free Encyclopedia.