Linear and multilinear algebra; matrix theory Definition

Invertible matrix

Builds on Square matrix Identity matrix Matrix multiplication

Definition

Let $R$ be a commutative ring and let $A$ be an $n \times n$ square matrix over $R$ . The matrix $A$ is invertible when there is an $n \times n$ matrix $B$ over $R$ such that $AB = BA = I_n,$ where $I_n$ is the identity matrix. The matrix $B$ is then unique; it is the inverse of $A$ , written $A^{-1}$ .

Remarks

Over a field, other names are nonsingular and nondegenerate. A matrix over $R$ is invertible if and only if its determinant is a unit of $R$ . Over a field, a one-sided inverse is two-sided, and $A$ is invertible if and only if $Ax = 0$ has only the solution $x = 0$ . The matrix $\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$ is not invertible over $\R$ , because its determinant is $0$ .

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Identity matrix object
Matrix definition
Matrix multiplication definition
Square matrix definition
Invertible matrix definition

Tag	`0TWTS88TWN`
URI	`m2://0TWTS88TWN`
Kind	definition
Writing	W1 stated
MSC	15A09
Defines	Invertible matrix
Uses	Square matrix, Identity matrix, Matrix multiplication

Invertible matrix

Definition

Remarks

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