# 2017年考研数二第07题解析

## 题目

$$A. \alpha_{1} + \alpha_{2}$$

$$B. \alpha_{2} + 2 \alpha_{3}$$

$$C. \alpha_{2} + \alpha_{3}$$

$$D. \alpha_{1} + 2 \alpha_{2}$$

## 解析

$$A(\alpha_{1} + \alpha_{2} + \alpha_{3}) =$$

$$A \alpha_{1} + A \alpha_{2} + A \alpha_{3}.$$

（将 $A$ 和 $\alpha$ 都看作 $1$ 阶矩阵就会发现上述运算过程是正确的。）

$$AP = P \begin{bmatrix} 0& 0& 0\\ 0& 1& 0\\ 0& 0& 2 \end{bmatrix} =$$

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

$$(0, \alpha_{2}, 2 \alpha_{3}).$$

$$AP = (A \alpha_{1}, A \alpha_{2}, A \alpha_{3}).$$

$$(A \alpha_{1}, A \alpha_{2}, A \alpha_{3}) = (0, \alpha_{2}, 2 \alpha_{3}).$$

$$A \alpha_{1} = 0;$$

$$A \alpha_{2} = \alpha_{2};$$

$$A \alpha_{3} = 2 \alpha_{3}$$

$$A \alpha_{1} + A \alpha_{2} + A \alpha_{3} =$$

$$0 + \alpha_{2} + 2 \alpha_{3} =$$

$$\alpha_{2} + 2 \alpha_{3}.$$

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

$$(\alpha_{1}, \alpha_{2}, \alpha_{3}) \begin{bmatrix} 1\\ 1\\ 1 \end{bmatrix} =$$

$$P \begin{bmatrix} 1\\ 1\\ 1 \end{bmatrix}.$$

$$A(\alpha_{1} + \alpha_{2} + \alpha_{3}) =$$

$$AP \begin{bmatrix} 1\\ 1\\ 1 \end{bmatrix}.$$

$$AP = P \begin{bmatrix} 0& 0& 0\\ 0& 1& 0\\ 0& 0& 2 \end{bmatrix}$$

$$A(\alpha_{1} + \alpha_{2} + \alpha_{3}) =$$

$$P \begin{bmatrix} 0& 0& 0\\ 0& 1& 0\\ 0& 0& 2 \end{bmatrix} \begin{bmatrix} 1\\ 1\\ 1 \end{bmatrix} =$$

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

$$(0, \alpha_{2}, 2 \alpha_{3}) \begin{bmatrix} 1\\ 1\\ 1 \end{bmatrix} =$$

$$\alpha_{2} + 2 \alpha_{3}.$$

$$A \alpha_{1} = \lambda_{1} \alpha_{1};$$

$$A \alpha_{2} = \lambda_{2} \alpha_{2};$$

$$A \alpha_{3} = \lambda_{3} \alpha_{3}.$$

$$\lambda_{1} =0;$$

$$\lambda_{2} =1;$$

$$\lambda_{3} =2;$$

$$A \alpha_{1} + A \alpha_{2} + A \alpha_{3} =$$

$$\lambda_{1} \alpha_{1} + \lambda_{2} \alpha_{2} + \lambda_{3} \alpha_{3} =$$

$$0 + \alpha_{2} + 2 \alpha_{3} =$$

$$\alpha_{2} + 2 \alpha_{3}.$$

EOF