问题
根据矩阵秩的性质，以下关于 $\textcolor{orange}{\mathrm{r}(\boldsymbol{A}, \boldsymbol{B})}$ 的 取 值 范 围 的选项中，正确的是哪个？选项
[A]. $\max \{\mathrm{r}(\boldsymbol{A}), \mathrm{r}(\boldsymbol{B})\}$ $<$ $\mathrm{r}(\boldsymbol{A}, \boldsymbol{B})$ $<$ $\mathrm{r}(\boldsymbol{A})$ $+$ $\mathrm{r}(\boldsymbol{B})$[B]. $\max \{\mathrm{r}(\boldsymbol{A}), \mathrm{r}(\boldsymbol{B})\}$ $\geqslant$ $\mathrm{r}(\boldsymbol{A}, \boldsymbol{B})$ $\geqslant$ $\mathrm{r}(\boldsymbol{A})$ $+$ $\mathrm{r}(\boldsymbol{B})$
[C]. $\min \{\mathrm{r}(\boldsymbol{A}), \mathrm{r}(\boldsymbol{B})\}$ $\leqslant$ $\mathrm{r}(\boldsymbol{A}, \boldsymbol{B})$ $\leqslant$ $\mathrm{r}(\boldsymbol{A})$ $+$ $\mathrm{r}(\boldsymbol{B})$
[D]. $\max \{\mathrm{r}(\boldsymbol{A}), \mathrm{r}(\boldsymbol{B})\}$ $\leqslant$ $\mathrm{r}(\boldsymbol{A}, \boldsymbol{B})$ $\leqslant$ $\mathrm{r}(\boldsymbol{A})$ $+$ $\mathrm{r}(\boldsymbol{B})$
$\textcolor{cyan}{\max} \{\mathrm{r}(\boldsymbol{A}), \mathrm{r}(\boldsymbol{B})\}$ $\textcolor{red}{\leqslant}$ $\textcolor{orange}{\mathrm{r}(\boldsymbol{A}, \boldsymbol{B})}$ $\textcolor{red}{\leqslant}$ $\mathrm{r}(\boldsymbol{A})$ $\textcolor{yellow}{+}$ $\mathrm{r}(\boldsymbol{B})$