# 看准题目所给条件，可以降低发生低级错误的可能性

## 二、解析

\begin{aligned} \left( 3 \boldsymbol{A} ^ {*} \right) ^ { – 1 } \\ \\ & = \left( 3 \begin{vmatrix} \boldsymbol{A} \end{vmatrix} \cdot \boldsymbol{A} ^{-1} \right) ^{-1} \\ \\ & = \frac{1}{3 \begin{vmatrix} \boldsymbol{\textcolor{orangered}{A} } \end{vmatrix}} \cdot \boldsymbol{ \textcolor{orangered}{A} } \end{aligned}

\begin{aligned} \begin{bmatrix} \textcolor{orangered}{1} & \textcolor{orangered}{0} & \textcolor{orangered}{0} & \vdots & \textcolor{pink}{1} & \textcolor{pink}{0} & \textcolor{pink}{0} \\ \textcolor{orangered}{2} & \textcolor{orangered}{1} & \textcolor{orangered}{0} & \vdots & \textcolor{pink}{0} & \textcolor{pink}{1} & \textcolor{pink}{0} \\ \textcolor{orangered}{3} & \textcolor{orangered}{2} & \textcolor{orangered}{1} & \vdots & \textcolor{pink}{0} & \textcolor{pink}{0} & \textcolor{pink}{1} \end{bmatrix} & \Rightarrow \\ \\ \begin{bmatrix} 1 & 0 & 0 & \vdots & 1 & 0 & 0 \\ 0 & 1 & 0 & \vdots & -2 & 1 & 0 \\ 3 & 2 & 1 & \vdots & 0 & 0 & 1 \end{bmatrix} & \Rightarrow \\ \\ \begin{bmatrix} \textcolor{pink}{1} & \textcolor{pink}{0} & \textcolor{pink}{0} & \vdots & \textcolor{yellow}{1} & \textcolor{yellow}{0} & \textcolor{yellow}{0} \\ \textcolor{pink}{0} & \textcolor{pink}{1} & \textcolor{pink}{0} & \vdots & \textcolor{yellow}{-2} & \textcolor{yellow}{1} & \textcolor{yellow}{0} \\ \textcolor{pink}{0} & \textcolor{pink}{0} & \textcolor{pink}{1} & \vdots & \textcolor{yellow}{1} & \textcolor{yellow}{-2} & \textcolor{yellow}{1} \end{bmatrix} & \Rightarrow \end{aligned}

$$\textcolor{yellow}{ \boldsymbol{A} = \begin{bmatrix} 1 & 0 & 0 \\ -2 & 1 & 0 \\ 1 & -2 & 1 \end{bmatrix} }$$

$$\textcolor{springgreen}{ \left( 3 \boldsymbol{A} ^ {*} \right) ^ { – 1 } = \frac { 1 } { 3 } \begin{bmatrix} 1 & 0 & 0 & \\ – 2 & 1 & 0 \\ 1 & – 2 & 1 \end{bmatrix} }$$

\begin{aligned} \left( 2 \boldsymbol {A} \right) ^ {*} \\ \\ & = \begin{vmatrix} 2 \boldsymbol{A} \end{vmatrix} \cdot \left( 2 \boldsymbol{A} \right) ^{-1} \\ \\ & = 2 ^{\boldsymbol{\textcolor{orangered}{3}}} \cdot \frac{1}{2} \boldsymbol{A} ^{-1} \\ \\ & = 4 \boldsymbol{A} ^{-1} \end{aligned}

$$\textcolor{springgreen}{ \left( 2 \boldsymbol {A} \right) ^ {*} = 4 \begin{bmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{bmatrix} }$$