设分块矩阵 $\boldsymbol{A}=\left(\begin{array}{ll}\boldsymbol{A}_{11} & \boldsymbol{A}_{12} \\ \boldsymbol{A}_{21} & \boldsymbol{A}_{22}\end{array}\right), B=\left(\begin{array}{ll}\boldsymbol{B}_{11} & \boldsymbol{B}_{12} \\ \boldsymbol{B}_{21} & \boldsymbol{B}_{22}\end{array}\right)$, 其中 $\boldsymbol{A}_{11}, \boldsymbol{B}_{11}$ 都是 $k$ 阶方阵, $\boldsymbol{A}_{22}, \boldsymbol{B}_{22}$ 都是 $n-k$ 阶方阵, 且满足 $\operatorname{r}(\boldsymbol{A})=\operatorname{r}\left(\boldsymbol{A}_{11}\right), \mathrm{r}(\boldsymbol{B})=$ $\mathrm{r}\left(\boldsymbol{B}_{22}\right)$. 求证:
$$
\left|\begin{array}{ll}
\boldsymbol{A}_{11} & \boldsymbol{B}_{12} \\
\boldsymbol{A}_{21} & \boldsymbol{B}_{22}
\end{array}\right| \cdot\left|\begin{array}{ll}
\boldsymbol{A}_{11} & \boldsymbol{A}_{12} \\
\boldsymbol{B}_{21} & \boldsymbol{B}_{22}
\end{array}\right|=|\boldsymbol{A}+\boldsymbol{B}| \cdot\left|\boldsymbol{A}_{11}\right| \cdot\left|\boldsymbol{B}_{22}\right| .
$$