Square root of determinant equals determinant of square root? Is it true that for a real-valued positive definite matrix $X$, $\sqrt{\det(X)} = \det(X^{1/2})$?
I know that this is indeed true for the $2 \times 2$ case but I haven't been able to find the answer or prove it myself for larger $X$.
Thanks a lot for any help.

I assume you mean by $X^{1/2}$ a matrix $Y$ such that $Y^2=X$, in which case
$$\begin{align}
\det Y^2=\det X & \implies (\det Y)^2=X\\
& \implies \det Y=\pm\sqrt{\det X)}
\end{align}$$
