In linear algebra, an invertible matrix (non-singular, non-degenerate or regular) is a square matrix that has an inverse. In other words, if a matrix is invertible, it can be multiplied by its inverse matrix to yield the identity matrix. Invertible matrices are the same size as their inverse. The inverse of a matrix represents the inverse operation, meaning if a matrix is applied to a particular vector, followed by applying the matrix's inverse, the result is the original vector.
Definition An n-by-n square matrix A is called invertible if there exists an n-by-n square matrix B such that A B = B A = I n , {\displaystyle \mathbf {AB} =\mathbf {BA} =\mathbf {I} _{n},} 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 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.
Basic idea A matrix can be viewed as a rule for transforming vectors. For example, a real
n × n {\displaystyle n\times n} matrix A {\displaystyle A} defines a linear transformation
x ↦ A x {\displaystyle x\mapsto Ax}
from the set R n {\displaystyle \mathbb {R} ^{n}} of n {\displaystyle n} -tuples of real numbers to itself. The matrix is invertible when this transformation can be undone by another linear transformation. In that case, there is a matrix A − 1 {\displaystyle A^{-1}} such that applying A {\displaystyle A} and then A − 1 {\displaystyle A^{-1}} , or applying A − 1 {\displaystyle A^{-1}} and then A {\displaystyle A} , returns every vector to where it started. Geometrically, an invertible matrix does not collapse space into a lower-dimensional set. It sends distinct vectors to distinct vectors and reaches every vector in the target space. For a real matrix, this is reflected by its determinant: an invertible matrix has nonzero determinant, while a matrix with determinant zero collapses volume to zero and is not invertible. Algebraically, invertibility means that the linear system
A x = b {\displaystyle Ax=b}
has a unique solution x {\displaystyle x} for every vector b {\displaystyle b} . Equivalently, the columns of A {\displaystyle A} form a basis of the vector space. These and several other equivalent characterizations are summarized by the invertible matrix theorem.
Examples
An invertiable matrix Consider the following 2-by-2 matrix:
A = ( − 1 3 2 1 − 1 ) {\displaystyle \mathbf {A} ={\begin{pmatrix}-1&{\tfrac {3}{2}}\\1&-1\end{pmatrix}}}
The matrix A {\displaystyle \mathbf {A} } is invertible, as it has inverse B = ( 2 3 2 2 ) , {\displaystyle \mathbf {B} ={\begin{pmatrix}2&3\\2&2\end{pmatrix}},} which can be confirmed by computing
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