Linear and multilinear algebra; matrix theory Proposition

Inverse of the inverse of a matrix

Statement

Let $A$ be an $n \times n$ invertible matrix with inverse $B$ . Then $B$ is also invertible, and its inverse is $A$ . In symbols, $(A^{-1})^{-1} = A$ .

Remarks

The proof is one line. Since $B$ is the inverse of $A$ , we have $AB = BA = I$ . These two equations say exactly that $A$ is an inverse of $B$ . By uniqueness of inverses, $A$ is the inverse of $B$ .

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Invertible matrix concept
Inverse of the inverse of a matrix statement

Tag	`TC3X7P2ESK`
URI	`m2://TC3X7P2ESK`
Kind	statement, proposition
Writing	W0 stub
MSC	15A09
Status	This copy of the site has no build data. Run `m2 status TC3X7P2ESK`.
About	Invertible matrix
Uses	Invertible matrix
Changed	4 November 2026

Inverse of the inverse of a matrix

Statement

Remarks

Reading path

Further reading

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