设 $\boldsymbol{A}, \boldsymbol{C}$ 为 $n$ 阶实对称阵, $\boldsymbol{B}$ 为 $n$ 阶实方阵, $\boldsymbol{D}=\operatorname{diag}\left\{d_1, d_2, \cdots\right.$, $\left.d_n\right\}, d_i>0(1 \leq i \leq n)$, 满足:
$$
\left|\begin{array}{cc}
\mathrm{i} \boldsymbol{A}+\boldsymbol{D} & \mathrm{i} \boldsymbol{B} \\
\boldsymbol{B}^{\prime} & \boldsymbol{C}
\end{array}\right|=0,
$$
其中 $\mathrm{i}=\sqrt{-1}$ 为虚数单位. 证明: $\left|B^2+C^2\right|=0$.