# Is the determinant of a complex matrix the complex conjugate of the determinant of it's complex conjugate matrix?

Apologies for the confusing title.

Suppose we have some square matrix $$A$$ with complex entries and it's conjugate matrix $$\bar{A}$$ whose entries are the complex conjugate of those in $$A$$.

Is it true that the determinant of one of them is the complex conjugate of the determinant of the other? It seems simple for small matrices but I don't know if it's true in general (or is there a transpose in there somewhere?). In other words does: $$\overline{\left(|A|\right)}={|\overline{A}|}$$

• You can delete your last sentence if you are satisfied with the edited formula. – user376343 Dec 3 '18 at 12:03

For $$z,w \in \mathbb{C}$$, we have $$\overline{zw} = \overline{z} \overline{w}$$ and $$\overline{z+w} = \overline{z} + \overline{w}$$ etc.
The determinant is obtained by performing various addition and and multiplication operations on its entries. Since complex conjugation can be done before or after these operations, your claim $$\overline{\det A} = \det \overline{A}$$ holds.