Determinant of a Unitary Matrix

[thepopasmurf]

Homework Statement

Let U be a unitary matrix. Show that for all vectors x that
|Ux| = |x|

Homework Equations

$$U^H=U^{-1}$$
$$|Ux|=|U||x|$$

The Attempt at a Solution

$$U^HU=I$$
$$|U^HU|=1$$
$$|U^T|^*|U|=1$$
$$(det(U))^2 = 1$$
so
det(U) = +/- 1

But that doesn't solve the question

 

[micromass]

First, convince yourself that $|a|=aa^H$. Then use this to calculate |Ux| and |x|.

 

[thepopasmurf]

Hang on, the vertical bars are standing for the magnitude, not the determinant? I need to do this all again.

 

[micromass]

Uh, well, if they mean determinant, then I have no idea what |x| would mean. You can't take the determinant of a vector...