# 利用好分块矩阵的性质，可以节省计算步骤

## 二、解析

\begin{aligned} \boldsymbol{A} \\ \\ = & \begin{bmatrix} \textcolor{orangered}{1} & \textcolor{orangered}{0} & \textcolor{orange}{0} & \textcolor{orange}{0} \\ \textcolor{orangered}{0} & \textcolor{orangered}{1} & \textcolor{orange}{0} & \textcolor{orange}{0} \\ \textcolor{springgreen}{1} & \textcolor{springgreen}{0} & \textcolor{orangered}{1} & \textcolor{orangered}{0} \\ \textcolor{springgreen}{2} & \textcolor{springgreen}{2} & \textcolor{orangered}{0} & \textcolor{orangered}{1} \end{bmatrix} \\ \\ = & \begin{bmatrix} \textcolor{orangered}{\boldsymbol{E}} & \textcolor{orange}{\boldsymbol{O}} \\ \textcolor{springgreen}{\boldsymbol{A}_{a}} & \textcolor{orangered}{\boldsymbol{E}} \end{bmatrix} \end{aligned}

\begin{aligned} \boldsymbol{B} \\ \\ = & \begin{bmatrix} \textcolor{tan}{1} & \textcolor{tan}{0} \\ \textcolor{tan}{1} & \textcolor{tan}{2} \\ \textcolor{red}{1} & \textcolor{red}{1} \\ \textcolor{red}{0} & \textcolor{red}{1} \end{bmatrix} \\ \\ = & \begin{bmatrix} \textcolor{tan}{\boldsymbol{B}_{b}} \\ \textcolor{red}{\boldsymbol{B}_{bb}} \\ \end{bmatrix} \end{aligned}

\begin{aligned} \boldsymbol{AB} \\ \\ = & \begin{bmatrix} \boldsymbol{E} & \boldsymbol{O} \\ \boldsymbol{A}_{a} & \boldsymbol{E} \end{bmatrix} \begin{bmatrix} \boldsymbol{B}_{b} \\ \boldsymbol{B}_{bb} \\ \end{bmatrix} \\ \\ = & \begin{bmatrix} \boldsymbol{B}_{b} \\ \boldsymbol{A}_{a} \boldsymbol{B}_{b} + \boldsymbol{B}_{bb} \end{bmatrix} \end{aligned}

\begin{aligned} \boldsymbol{A}_{a} \boldsymbol{B}_{b} \\ \\ = & \begin{bmatrix} 1 & 0 \\ 2 & 2 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 1 & 2 \end{bmatrix} \\ \\ = & \begin{bmatrix} 1 & 0 \\ 4 & 4 \end{bmatrix} \end{aligned}

$$\boldsymbol{A}_{a} \boldsymbol{B}_{b} + \boldsymbol{B}_{bb} = \begin{bmatrix} \textcolor{magenta}{2} & \textcolor{magenta}{1} \\ \textcolor{magenta}{4} & \textcolor{magenta}{5} \end{bmatrix}$$

$$\textcolor{springgreen}{\boldsymbol{AB}} = \begin{bmatrix} \textcolor{tan}{1} & \textcolor{tan}{0} \\ \textcolor{tan}{1} & \textcolor{tan}{2} \\ \textcolor{magenta}{2} & \textcolor{magenta}{1} \\ \textcolor{magenta}{4} & \textcolor{magenta}{5} \end{bmatrix}$$