Question

If u is a square matrix such that u^tu = i what can we say about det(u)

Answer 1

Recall that

$\det(\mathbf{AB})=\det\mathbf A\det\mathbf B$

$\det\mathbf A=\det\mathbf A^\top$

for any square matrices $\mathbf A,\mathbf B$. So if

$\mathbf U^\top\mathbf U=\mathbf I$, and $\det\mathbf I=1$, we have

$\det(\mathbf U^\top\mathbf U)=(\det\mathbf U)^2=1\implies\det\mathbf U=\pm1$