# 2012年考研数二第08题解析

## 题目

$$A. \begin{pmatrix} 1 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 1 \end{pmatrix}$$

$$B. \begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 2 \end{pmatrix}$$

$$C. \begin{pmatrix} 2 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 2 \end{pmatrix}$$

$$D. \begin{pmatrix} 2 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 1 \end{pmatrix}$$

## 解析

### 方法一

$$P=E.$$

$$P^{-1} = E.$$

$$P^{-1} A P = EAE = A =$$

$$\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 2 \end{pmatrix}.$$

$$P=(\alpha_{1}, \alpha_{2}, \alpha_{3});$$

$$Q=(\alpha_{1} + \alpha_{2}, \alpha_{2}, \alpha_{3}).$$

$$Q = \begin{pmatrix} 1 & 0 & 0\\ 1 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}$$

$$Q^{-1}= \begin{pmatrix} 1 & 0 & 0\\ -1 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}$$

$$Q^{-1}AQ= \begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 2 \end{pmatrix}.$$

### 方法二

$$P=(\alpha_{1}, \alpha_{2}, \alpha_{3});$$

$$Q=(\alpha_{1} + \alpha_{2}, \alpha_{2}, \alpha_{3}).$$

$$Q=$$

$$(\alpha_{1}, \alpha_{2}, \alpha_{3}) \begin{pmatrix} 1 & 0 & 0\\ 1 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}$$

$$K= \begin{pmatrix} 1 & 0 & 0\\ 1 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}$$

$$Q=PK.$$

$$Q^{-1} = (PK)^{-1} = K^{-1}P^{-1}.$$

$$K^{-1} =$$

$$\begin{pmatrix} 1 & 0 & 0\\ -1 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}$$

$$Q^{-1}AQ =$$

$$K^{-1}(P^{-1}AP)K =$$

$$\begin{pmatrix} 1 & 0 & 0\\ -1 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 2 \end{pmatrix} \begin{pmatrix} 1 & 0 & 0\\ 1 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix} =$$

$$\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 2 \end{pmatrix}.$$

EOF