If A is unitary and $\det (A^H) = \det(A^*)$ and $\det(A) \det(A^*) = \det(A)^2$ Why can’t I say that $\det(A^H) = \pm\det(A)$

Question

this came up on a Homework I have.
I had to prove that the absolute value of the determinant of a Unitary Matrix is 1.

So because $\det (A^H) = \det(A^*)$
and $\det(A) \det(A^*) = \det(A)^2$
as well as that for a Unitary Matrix $A*A^H=I$ that $\det(A^*) = \pm \det(A)$

This was my wrong step.

And the correct one confuses me even more.
It's $\det (A^H) = \det(conjugate(A))$

$\det(A) \det(A^*) = \det(A)^2$ then

$\det(A)^2 = I = 1$

Which looks like it's saying the same to me. Thank you to anyone who may shed some light on this for me, it's much appreciated.

Answer 1

The product of a complex number with its conjugate is equal to its squared modulus. Therefore $\det(A)\det(\overline{A})$ in general is equal to $|\det(A)|^2$, not $\det(A)^2$.