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