Invertible matrix

In linear algebra, 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

\mathbf {AB} =\mathbf {BA} =\mathbf {I} _{n},

where

I n

denotes the

n

-by-

n

identity matrix and the multiplication used is ordinary matrix multiplication.^[1] 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.^{[citation needed]}

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 $m \neq n$ , 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 $n$ , ( $n \leq m$ ), then $A$ has a left inverse, an $n$ -by- $m$ matrix $B$ such that $BA = I n$ . If $A$ has rank $m$ ( $m \leq n$ ), then it has a right inverse, an $n$ -by- $m$ matrix $B$ such that $AB = I m$ .

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 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 $n \times n$ 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 $GL n (R)$ .

^ Axler, Sheldon (18 December 2014). Linear Algebra Done Right. Undergraduate Texts in Mathematics (3rd ed.). Springer Publishing (published 2015). p. 296. ISBN 978-3-319-11079-0.

[1] Axler, Sheldon (18 December 2014). Linear Algebra Done Right. Undergraduate Texts in Mathematics (3rd ed.). Springer Publishing (published 2015). p. 296. ISBN 978-3-319-11079-0.

[1]