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.

Call $Y = x^{\frac{1}{2}}$. Then $\det X = \det (Y^2) = (\det Y)^2$, so $\sqrt{\det X} = |\det Y|$. If $Y$ is positive definite, then $|\det Y| = \det Y$, and so we are done.
