# Determinant of Inverse Matrix

## Theorem

Let $K$ be a field whose zero is $0_K$ and whose unity is $1_K$.

Let $\mathbf A$ be an invertible matrix of order $n$ over $K$.

Then the determinant of its inverse is given by:

$\map \det {\mathbf A^{-1} } = \dfrac {1_K} {\map \det {\mathbf A} }$

## Proof

By definition of inverse matrix:

$\mathbf A \mathbf A^{-1} = \mathbf I_n$

where $\mathbf I_n$ is the unit matrix.

$\map \det {\mathbf I_n} = 1_K$
$\map \det {\mathbf A^{-1} } \map \det {\mathbf A} = \map \det {\mathbf A^{-1} \mathbf A}$

Hence the result.

$\blacksquare$