# 利用“对称初等变换”求解合同矩阵中的可逆矩阵 C

## 二、解析

\begin{aligned} & \begin{bmatrix} 0 & 1 & 1 & \textcolor{gray}{|} & 1 & 0 & 0 \\ 1 & 3 & 1 & \textcolor{gray}{|} & 0 & 1 & 0 \\ 1 & 1 & 0 & \textcolor{gray}{|} & 0 & 0 & 1 \end{bmatrix} \\ \\ \xRightarrow{r_{1} \leftrightarrow r_{2}} & \begin{bmatrix} \textcolor{orangered}{1} & \textcolor{orangered}{3} & \textcolor{orangered}{1} & \textcolor{gray}{|} & \textcolor{white}{\colorbox{green}{0}} & \textcolor{white}{\colorbox{green}{1}} & \textcolor{white}{\colorbox{green}{0}} \\ \textcolor{orangered}{0} & \textcolor{orangered}{1} & \textcolor{orangered}{1} & \textcolor{gray}{|} & \textcolor{white}{\colorbox{green}{1}} & \textcolor{white}{\colorbox{green}{0}} & \textcolor{white}{\colorbox{green}{0}} \\ \textcolor{orangered}{1} & \textcolor{orangered}{1} & \textcolor{orangered}{0} & \textcolor{gray}{|} & \textcolor{white}{\colorbox{green}{0}} & \textcolor{white}{\colorbox{green}{0}} & \textcolor{white}{\colorbox{green}{1}} \end{bmatrix} \end{aligned}

\begin{aligned} & \begin{bmatrix} \textcolor{orangered}{1} & \textcolor{orangered}{3} & \textcolor{orangered}{1} \\ \textcolor{orangered}{0} & \textcolor{orangered}{1} & \textcolor{orangered}{1} \\ \textcolor{orangered}{1} & \textcolor{orangered}{1} & \textcolor{orangered}{0} \\ \textcolor{gray}{-} & \textcolor{gray}{-} & \textcolor{gray}{-} \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \\ \\ \xRightarrow{c_{1} \leftrightarrow c_{2}} & \begin{bmatrix} \textcolor{springgreen}{3} & \textcolor{springgreen}{1} & \textcolor{springgreen}{1} \\ \textcolor{springgreen}{1} & \textcolor{springgreen}{0} & \textcolor{springgreen}{1} \\ \textcolor{springgreen}{1} & \textcolor{springgreen}{1} & \textcolor{springgreen}{0} \\ \textcolor{gray}{-} & \textcolor{gray}{-} & \textcolor{gray}{-} \\ \textcolor{brown}{\colorbox{yellow}{0}} & \textcolor{brown}{\colorbox{yellow}{1}} & \textcolor{brown}{\colorbox{yellow}{0}} \\ \textcolor{brown}{\colorbox{yellow}{1}} & \textcolor{brown}{\colorbox{yellow}{0}} & \textcolor{brown}{\colorbox{yellow}{0}} \\ \textcolor{brown}{\colorbox{yellow}{0}} & \textcolor{brown}{\colorbox{yellow}{0}} & \textcolor{brown}{\colorbox{yellow}{1}} \end{bmatrix} \end{aligned}

$$\begin{bmatrix} \textcolor{springgreen}{3} & \textcolor{springgreen}{1} & \textcolor{springgreen}{1} \\ \textcolor{springgreen}{1} & \textcolor{springgreen}{0} & \textcolor{springgreen}{1} \\ \textcolor{springgreen}{1} & \textcolor{springgreen}{1} & \textcolor{springgreen}{0} \end{bmatrix} = \boldsymbol{B}$$

$$\boldsymbol{C} ^{\top} = \begin{bmatrix} \textcolor{white}{\colorbox{green}{0}} & \textcolor{white}{\colorbox{green}{1}} & \textcolor{white}{\colorbox{green}{0}} \\ \textcolor{white}{\colorbox{green}{1}} & \textcolor{white}{\colorbox{green}{0}} & \textcolor{white}{\colorbox{green}{0}} \\ \textcolor{white}{\colorbox{green}{0}} & \textcolor{white}{\colorbox{green}{0}} & \textcolor{white}{\colorbox{green}{1}} \end{bmatrix}$$

$$\boldsymbol{C} = \begin{bmatrix} \textcolor{brown}{\colorbox{yellow}{0}} & \textcolor{brown}{\colorbox{yellow}{1}} & \textcolor{brown}{\colorbox{yellow}{0}} \\ \textcolor{brown}{\colorbox{yellow}{1}} & \textcolor{brown}{\colorbox{yellow}{0}} & \textcolor{brown}{\colorbox{yellow}{0}} \\ \textcolor{brown}{\colorbox{yellow}{0}} & \textcolor{brown}{\colorbox{yellow}{0}} & \textcolor{brown}{\colorbox{yellow}{1}} \end{bmatrix}$$