设 $\boldsymbol{A}$ 为 $n$ 阶正定实对称阵, $\boldsymbol{B}, \boldsymbol{C}$ 为 $n$ 阶半正定实对称阵, 使得 $\boldsymbol{B} \boldsymbol{A}^{-1} \boldsymbol{C}$ 为对称阵. 证明:
$
|\boldsymbol{A}| \cdot|\boldsymbol{A}+\boldsymbol{B}+\boldsymbol{C}| \leq|\boldsymbol{A}+\boldsymbol{B}| \cdot|\boldsymbol{A}+\boldsymbol{C}|,
$
并求等号成立的充要条件.