# 单位矩阵可以用来记录初等变换

## 二、正文

### 单位矩阵对一般矩阵的作用

$$\textcolor{orange}{ \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} } \begin{bmatrix} \textcolor{white}{\colorbox{green}{1}} & \textcolor{white}{\colorbox{green}{2}} \\ \textcolor{brown}{\colorbox{yellow}{3}} & \textcolor{brown}{\colorbox{yellow}{4}} \end{bmatrix} = \begin{bmatrix} \textcolor{brown}{\colorbox{yellow}{3}} & \textcolor{brown}{\colorbox{yellow}{4}} \\ \textcolor{white}{\colorbox{green}{1}} & \textcolor{white}{\colorbox{green}{2}} \end{bmatrix}$$

$$\begin{bmatrix} \textcolor{white}{\colorbox{green}{1}} & \textcolor{brown}{\colorbox{yellow}{2}} \\ \textcolor{white}{\colorbox{green}{3}} & \textcolor{brown}{\colorbox{yellow}{4}} \end{bmatrix} \textcolor{orange}{ \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} } = \begin{bmatrix} \textcolor{brown}{\colorbox{yellow}{2}} & \textcolor{white}{\colorbox{green}{1}} \\ \textcolor{brown}{\colorbox{yellow}{4}} & \textcolor{white}{\colorbox{green}{3}} \end{bmatrix}$$

### 单位矩阵的“记录”功能

$$\begin{bmatrix} \textcolor{white}{\colorbox{green}{1}} & \textcolor{white}{\colorbox{green}{2}} & \textcolor{gray}{|} & 1 & 0 \\ \textcolor{brown}{\colorbox{yellow}{3}} & \textcolor{brown}{\colorbox{yellow}{4}} & \textcolor{gray}{|} & 0 & 1 \end{bmatrix} \Rightarrow \begin{bmatrix} \textcolor{brown}{\colorbox{yellow}{3}} & \textcolor{brown}{\colorbox{yellow}{4}} & \textcolor{gray}{|} & 0 & 1 \\ \textcolor{white}{\colorbox{green}{1}} & \textcolor{white}{\colorbox{green}{2}} & \textcolor{gray}{|} & 1 & 0 \end{bmatrix}$$

$$\begin{bmatrix} \textcolor{white}{\colorbox{green}{1}} & \textcolor{white}{\colorbox{green}{2}} \\ \textcolor{brown}{\colorbox{yellow}{3}} & \textcolor{brown}{\colorbox{yellow}{4}} \\ \textcolor{gray}{-} & \textcolor{gray}{-} \\ 1 & 0 \\ 0 & 1 \end{bmatrix} \Rightarrow \begin{bmatrix} \textcolor{white}{\colorbox{green}{2}} & \textcolor{white}{\colorbox{green}{1}} \\ \textcolor{brown}{\colorbox{yellow}{4}} & \textcolor{brown}{\colorbox{yellow}{3}} \\ \textcolor{gray}{-} & \textcolor{gray}{-} \\ 0 & 1 \\ 1 & 0 \end{bmatrix}$$