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 n-by-n square matrix A is called invertible (also nonsingular, nondegenerate or rarely regular)
if there exists an n-by-n square 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
multiplicative inverse of A, denoted by A−1. 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. 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, (n ≤ m), then A has a left inverse, an n-by-m matrix B such that BA = In. If A has
rank m(m ≤ n), then it has a right inverse, an n-by-m matrix B such that AB = Im.
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 n × n invertible matrices together with the operation of matrix multiplication and
entries from ring R form a group, the general linear group of degree n, denoted GLn(R).
This article is a derivative work of the creative commons share alike with attribution in [3].
References
[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.
