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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: $$\overline{\left(|A|\right)}={|\overline{A}|}$$

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  • $\begingroup$ You can delete your last sentence if you are satisfied with the edited formula. $\endgroup$
    – user376343
    Commented Dec 3, 2018 at 12:03

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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.

Regarding your last sentence, note also that transposing a matrix does not change its determinant.

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  • $\begingroup$ Thanks, exactly what I needed, I'll accept after the allowed time has passed $\endgroup$
    – R. Rankin
    Commented Dec 3, 2018 at 4:34

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