The determinant of an inverse matrix of A is equal to the reciprocal of determinant of A :
$$\mathrm{det }(A^{-1})=(\mathrm{det } A)^{-1}$$
Proof
By the property of the determinant, we have
$$\mathrm{det }(AB)=\mathrm{det }A\ \mathrm{det }B$$
Then we have
$$1=\mathrm{det }\ I=\mathrm{det } (AA^{-1})=\mathrm{det }A\ \mathrm{det }A^{-1}$$
where \(I\) is an identity matrix.
Therefore, we obtain
$$\mathrm{det }(A^{-1})=(\mathrm{det } A)^{-1}=\frac{1}{\mathrm{det }A}$$