I'm having a difficult time determining if the following function is convex
$$f(X) = \log {\rm det}(X^T A X)$$
where $A \in \mathbb{R}^{r \times r}$ is a symmetric positive definite matrix and $X \in \mathbb{R}^{r \times u}$ with $u < r$ and $X'X = I$. I've worked on find the second derivative with respect to $X$, but am not very confident in my solution. Any suggests on steps to proceed would be appreciated.
EDIT: Because the domain of $X$ is nonconvex, the function cannot be convex, as pointed out by @RobertIsrael.