Consider the following log-determinant function
$$f\left( {\bf{X}} \right) = \log |{\bf{A}} + {\bf{B}\bf{X}}^T + {\bf{X}}{{\bf{X}}^T}|$$
where ${\bf{X}} \in {\mathbb R^{K \times K'}}$ and ${\bf{A}}$ and ${\bf{B}}$ are some matrices with proper dimensions. We assume $K' > K$. Therefore, ${\bf{X}}{{\bf{X}}^T}$ is a positive definite matrix and function $f\left( {\bf{X}} \right)$ is well-defined. The goal is to check whether $f \left( {\bf{X}} \right)$ is convex or not. Any hint for this?