  This note provides another proof for the {\em convexity} ({\em strict
convexity}) of $\log \det ( I + KX^{-1} )$ over the positive definite cone for
any given positive semidefinite matrix $K \succeq 0$ (positive definite matrix
$K \succ 0$) and the {\em strictly convexity} of $\log \det (K + X^{-1})$ over
the positive definite cone for any given $K \succeq 0$. Equivalent optimization
representation with linear matrix inequalities (LMIs) for the functions $\log
\det ( I + KX^{-1} )$ and $\log \det (K + X^{-1})$ are presented. Their
optimization representations with LMI constraints can be particularly useful
for some related synthetic design problems.
