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:
 * $\det \left({\mathbf A^{-1}}\right) = \dfrac {1_K} {\det \left({\mathbf A}\right)}$

Proof
We have by definition that $\mathbf A \mathbf A^{-1} = \mathbf I_n$ where $\mathbf I_n$ is the identity matrix.

We also have by Determinant of Identity Matrix that $\det \left({\mathbf I_n}\right) = 1_K$.

We also have by Determinant of Matrix Product that:
 * $\det \left({\mathbf A^{-1}}\right) \det \left({\mathbf A}\right) = \det \left({\mathbf A^{-1} \mathbf A}\right)$

Hence the result.