设 $n$ 阶方阵 $\boldsymbol{A}, \boldsymbol{B}$ 满足 $\boldsymbol{A B}=\boldsymbol{B} \boldsymbol{A}$ 且 $\mathrm{r}(\boldsymbol{A}) \geq n-1$. 证明:
$\mathrm{r}\left(\boldsymbol{A}^2\right)+\mathrm{r}\left(\boldsymbol{B}^2\right) \geq 2 \mathrm{r}(\boldsymbol{A} \boldsymbol{B}) $