# 矩阵乘法的次幂是不能放到括号里面的：即便他们相乘得单位矩阵

## 二、解析

\begin{aligned} & (\boldsymbol{AB}) ^{2} = \boldsymbol{E} \\ \\ \Leftrightarrow & \boldsymbol{A} (\textcolor{magenta}{\boldsymbol{BAB}}) = \boldsymbol{E} \\ \\ \Leftrightarrow & \boldsymbol{A} \textcolor{magenta}{\boldsymbol{A} ^{-1}} = \boldsymbol{E} \end{aligned}

$$\textcolor{springgreen}{ \boldsymbol{BAB} = \boldsymbol{A} ^{-1} }$$

\begin{aligned} & (\boldsymbol{AB}) ^{2} = \boldsymbol{E} \\ \\ \Leftrightarrow & (\textcolor{magenta}{\boldsymbol{ABA}}) \boldsymbol{B} = \boldsymbol{E} \\ \\ \Leftrightarrow & \textcolor{magenta}{\boldsymbol{B} ^{-1}} \boldsymbol{B} = \boldsymbol{E} \end{aligned}

$$\textcolor{springgreen}{ \boldsymbol{ABA} = \boldsymbol{B} ^{-1} }$$

$$\boldsymbol{AB} = \boldsymbol{E}$$

$$\boldsymbol{AB} = (-1)^{\alpha} \boldsymbol{E}$$

$$\boldsymbol{BA} = \boldsymbol{E}$$

$$\boldsymbol{BA} = (-1) ^{\alpha} \boldsymbol{E}$$

$$\textcolor{springgreen}{ (\boldsymbol{BA}) ^{2} = \boldsymbol{E} }$$

\begin{aligned} & (\boldsymbol{AB}) ^{2} = \boldsymbol{E} \\ \\ \Leftrightarrow & (\boldsymbol{ABAB}) = \boldsymbol{E} \\ \\ \Leftrightarrow & \textcolor{tan}{\boldsymbol{B}} (\boldsymbol{ABAB}) \textcolor{pink}{\boldsymbol{A}} = \textcolor{tan}{\boldsymbol{B}} \boldsymbol{E} \textcolor{pink}{\boldsymbol{A}} \\ \\ \Leftrightarrow & (\boldsymbol{BA}) (\boldsymbol{BA}) (\boldsymbol{BA}) = (\boldsymbol{BA}) \boldsymbol{E} \\ \\ \Leftrightarrow & (\boldsymbol{BA}) (\boldsymbol{BA}) = \boldsymbol{E} \\ \\ \Leftrightarrow & \textcolor{springgreen}{(\boldsymbol{BA}) ^{2} = \boldsymbol{E}} \end{aligned}

\begin{aligned} & (\boldsymbol{AB}) ^{2} = \boldsymbol{E} \\ \\ \Leftrightarrow & \boldsymbol{A} (\boldsymbol{BA}) \boldsymbol{B} = \boldsymbol{E} \\ \\ \Leftrightarrow & \textcolor{pink}{\boldsymbol{A}^{-1}} \boldsymbol{A} (\boldsymbol{BA}) \boldsymbol{B} \textcolor{tan}{\boldsymbol{B} ^{-1}} = \textcolor{pink}{\boldsymbol{A}^{-1}} \boldsymbol{E} \textcolor{tan}{\boldsymbol{B} ^{-1}} \\ \\ \Leftrightarrow & \boldsymbol{BA} = \textcolor{pink}{\boldsymbol{A}^{-1}} \textcolor{tan}{\boldsymbol{B} ^{-1}} \\ \\ \Leftrightarrow & (\boldsymbol{BA}) ^{2} = \boldsymbol{BA} (\boldsymbol{A} ^{-1} \boldsymbol{B} ^{-1}) \\ \\ \Leftrightarrow & (\boldsymbol{BA}) ^{2} = \boldsymbol{B} \boldsymbol{E} \boldsymbol{B} ^{-1} \\ \\ \Leftrightarrow & \textcolor{springgreen}{(\boldsymbol{BA}) ^{2} = \boldsymbol{E}} \end{aligned}

1. 如果 $\boldsymbol{AB}$ $=$ $\boldsymbol{E}$, 则：

\begin{aligned} & \textcolor{magenta}{\boldsymbol{AB}} = \textcolor{magenta}{\boldsymbol{E}} \\ \\ \Leftrightarrow & \boldsymbol{A} (\textcolor{magenta}{\boldsymbol{AB}}) \boldsymbol{B} = \boldsymbol{A} \textcolor{magenta}{\boldsymbol{E}} \boldsymbol{B} \\ \\ \Leftrightarrow & \boldsymbol{AABB} = \textcolor{yellow}{\boldsymbol{AB}} \\ \\ \Leftrightarrow & \boldsymbol{AABB} = \textcolor{yellow}{\boldsymbol{E}} \\ \\ \Leftrightarrow & \textcolor{springgreen}{\boldsymbol{A} ^{2} \boldsymbol{B} ^{2} = \boldsymbol{E}} \end{aligned}

1. $\boldsymbol{AB}$ $=$ $(-1) ^{\alpha} \boldsymbol{E}$, 则：

\begin{aligned} & \textcolor{magenta}{\boldsymbol{AB}} = \textcolor{magenta}{(-1)^{\alpha} \boldsymbol{E}} \\ \\ \Leftrightarrow & \boldsymbol{A} (\textcolor{magenta}{\boldsymbol{AB}}) \boldsymbol{B} = \boldsymbol{A} (\textcolor{magenta}{(-1)^{\alpha} \boldsymbol{E}}) \boldsymbol{B} \\ \\ \Leftrightarrow & \boldsymbol{AABB} = \textcolor{yellow}{(-1)^{\alpha} \boldsymbol{AB}} \\ \\ \Leftrightarrow & \boldsymbol{AABB} = \textcolor{yellow}{(-1)^{2\alpha} \boldsymbol{E}} \\ \\ \Leftrightarrow & \textcolor{springgreen}{\boldsymbol{A} ^{2} \boldsymbol{B} ^{2} = (-1)^{2\alpha} \boldsymbol{E}} \end{aligned}

$$\boldsymbol{A} ^{2} \boldsymbol{B} ^{2} = (-1)^{2\alpha} \boldsymbol{E} \textcolor{orangered}{\neq \boldsymbol{E}}$$

\begin{aligned} \boldsymbol{A} & = \begin{bmatrix} 1 & -1 \\ 0 & 1 \end{bmatrix} \\ \\ \boldsymbol{B} & = \begin{bmatrix} 1 & -1 \\ 0 & -1 \end{bmatrix} \end{aligned}

$$\boldsymbol{AB} = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$$

\begin{aligned} & \boldsymbol{AB} \\ \\ & = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \\ \\ & = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \\ \\ & = \boldsymbol{E} \end{aligned}

\begin{aligned} & \begin{cases} \boldsymbol{A} ^{2} & = \begin{bmatrix} 1 & -2 \\ 0 & 1 \end{bmatrix} \\ \\ \boldsymbol{B} ^{2} & = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \end{cases} \\ \\ \Rightarrow & \boldsymbol{A} ^{2} \boldsymbol{B} ^{2} \\ \\ = & \begin{bmatrix} 1 & -2 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \\ \\ = & \begin{bmatrix} 1 & -2 \\ 0 & 1 \end{bmatrix} \boldsymbol{E} \\ \\ = & \boldsymbol{A} ^{2} \boldsymbol{E} \\ \\ = & \boldsymbol{A} ^{2} \textcolor{orangered}{\neq \boldsymbol{E}} \end{aligned}