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.

If you know that $\det(AB)=\det(A)\det(B)$, then this is immediate, taking $A=B=X^{1/2}$. If you don't know that, there are several proofs suggested here.
