Linear and multilinear algebra; matrix theory Proposition

Determinant of the inverse is the reciprocal of the determinant

Statement

Let $A$ be a square matrix with entries in a field, for example the real numbers, and suppose that $A$ is invertible. Then $\det A \neq 0$ , and the determinant of the inverse is the reciprocal of the determinant of $A$ : $\det(A^{-1}) = \frac{1}{\det A}.$

Remarks

The determinant is multiplicative, and $A A^{-1} = I$ has determinant $1$ . So $\det A \cdot \det(A^{-1}) = 1$ , and the claim follows.

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Determinant concept
Invertible matrix concept
Determinant of the inverse is the reciprocal of the determinant statement

Tag	`A6JV1E99V4`
URI	`m2://A6JV1E99V4`
Kind	statement, proposition
Writing	W0 stub
MSC	15A15
Status	This copy of the site has no build data. Run `m2 status A6JV1E99V4`.
Uses	Invertible matrix, Determinant
Changed	4 November 2026

Determinant of the inverse is the reciprocal of the determinant

Statement

Remarks

Reading path

Further reading

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