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

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.