kaoyan3basic 线性代数 第345题
📝 题目
### 第345题 345 设 $\boldsymbol{A}=\left[\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}, \boldsymbol{\alpha}_{3}\right]$ 是三阶矩阵，则下列行列式中等于 $|\boldsymbol{A}|$ 的是 （A）$\left|\boldsymbol{\alpha}_{1}-\boldsymbol{\alpha}_{2}, \boldsymbol{\alpha}_{2}-\boldsymbol{\alpha}_{3}, \boldsymbol{\alpha}_{3}-\boldsymbol{\alpha}_{1}\right|$ ． （B）$\left|\boldsymbol{\alpha}_{1}+\boldsymbol{\alpha}_{2}, \boldsymbol{\alpha}_{2}+\boldsymbol{\alpha}_{3}, \boldsymbol{\alpha}_{3}+\boldsymbol{\alpha}_{1}\right|$. （C）$\left|\boldsymbol{\alpha}_{1}+2 \boldsymbol{\alpha}_{2}, \boldsymbol{\alpha}_{3}, \boldsymbol{\alpha}_{1}+\boldsymbol{\alpha}_{2}\right|$ ． （D）$\left|\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}+\boldsymbol{\alpha}_{3}, \boldsymbol{\alpha}_{1}+\boldsymbol{\alpha}_{2}\right|$.
💡 答案解析
**答案**：C **解析**：步骤1：选项A：$|\alpha_1-\alpha_2, \alpha_2-\alpha_3, \alpha_3-\alpha_1|$，三列相加为0，行列式为0。步骤2：选项B：$|\alpha_1+\alpha_2, \alpha_2+\alpha_3, \alpha_3+\alpha_1|$，三列相加为$2(\alpha_1+\alpha_2+\alpha_3)$，行列式值为$2|\alpha_1+\alpha_2+\alpha_3, \alpha_2+\alpha_3, \alpha_3+\alpha_1|$，化简得$2|\alpha_1, \alpha_2, \alpha_3|$，不等于$|A|$。步骤3：选项C：$|\alpha_1+2\alpha_2, \alpha_3, \alpha_1+\alpha_2|$，第三列减第一列得$|\alpha_1+2\alpha_2, \alpha_3, -\alpha_2|$，第一列加2倍第三列得$|\alpha_1, \alpha_3, -\alpha_2|$，交换第二、三列得$-|\alpha_1, -\alpha_2, \alpha_3|=|\alpha_1, \alpha_2, \alpha_3|=|A|$。步骤4：选项D：$|\alpha_1, \alpha_2+\alpha_3, \alpha_1+\alpha_2|$，第三列减第一列得$|\alpha_1, \alpha_2+\alpha_3, \alpha_2|$，第二列减第三列得$|\alpha_1, \alpha_3, \alpha_2|=-|A|$。 **难度**：★★★☆☆