TITLE: Prove that determinant complex conjugate is complex conjugate of determinant
QUESTION [15 upvotes]: Prove that determinant complex conjugate is complex conjugate of determinant.
I know that there is an induction proof for this. However, I am wondering if there is a more elegant and simple solution that shows this fact. Thanks.

REPLY [13 votes]: This can easily be shown by induction also. If you assume the truth for an $n\times n$ matrix, then as the $(n+1)\times (n+1)$ matrix determinant is the sum of $n\times n$ determinants (a la cofactor expansion), you're essentially done.
The base case (and to illustrate how it works for the general case):
$$\overline{\det\left(\begin{matrix}a & b\\ c& d\end{matrix}\right)} = \overline{a\det(d)-c\det(b)} = \overline{a}\overline d-\overline c\overline b = \overline a\det(\overline d)-\overline c\det(\overline b ) =\det\left(\begin{matrix}\overline a & \overline b\\ \overline c& \overline d\end{matrix}\right)$$