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

  • $\begingroup$ You can delete your last sentence if you are satisfied with the edited formula. $\endgroup$ – 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.

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

  • $\begingroup$ Thanks, exactly what I needed, I'll accept after the allowed time has passed $\endgroup$ – R. Rankin Dec 3 '18 at 4:34

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.