## 一、题目

$$\begin{cases} \boldsymbol{A} = \frac{1}{3} (\boldsymbol{B} + \boldsymbol{E}) \\ \\ \boldsymbol{A} ^{2} = \boldsymbol{A} \end{cases}$$

$$\boldsymbol{B} = ?$$

## 一、前言

$$D _{ n } = \begin{vmatrix} 1 & 1 & 1 & \cdots & 1 \\ x _{ 1 } & x _{ 2 } & x _{ 3 } & \cdots & x _{ n } \\ x _{ 1 } ^ { 2 } & x _{ 2 } ^ { 2 } & x _{ 3 } ^ { 2 } & \cdots & x _{ n } ^ { 2 } \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ x _{ 1 } ^ { n – 1 } & x _{ 2 } ^ { n – 1 } & x _{ 3 } ^ { n – 1 } & \cdots & x _{ n } ^ { n – 1 } \end{vmatrix}$$

$$D _{ n } = \prod _{ 1 \leqslant j < i \leqslant n } \left( x _{ i } – x _{ j } \right)$$

## 一、题目

\begin{aligned} \boldsymbol{A} & = \begin{bmatrix} 0 & 1 & 0 \\ – 1 & -1 & 0 \\ 0 & 0 & 2 \end{bmatrix} \\ \\ \boldsymbol{B} & = \begin{bmatrix} 1 & – 1 \\ 2 & 0 \\ 3 & 1 \end{bmatrix} \end{aligned}

## 一、前言

$$\left|\begin{matrix} a_{11} & a_{12} & \cdots & a_{1⁢⁢⁢n} \\ a_{21} & a_{22} & \cdots & a_{2⁢⁢⁢n} \\ \vdots & \vdots & & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{m}\end{matrix}\right| = \textcolor{yellow}{\sum _{j_{1} j_{2} \cdots j_{n}}} \textcolor{springgreen}{\left(−1\right)^{\tau \left(j_{1}j_{2} \cdots j_{n}\right)}} \textcolor{pink}{a_{1j_{1}}a_{2j_{2}} \cdots a_{n}}$$

## 一、前言

$$\boldsymbol{AB} = \boldsymbol{O}$$