Invertible matrix

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.