题号:1876    题型:单选题    来源:2012年全国硕士研究生招生考试试题
设 $\boldsymbol{\alpha}_{1}=\left(\begin{array}{l}0 \\ 0 \\ c_{1}\end{array}\right), \boldsymbol{\alpha}_{2}=\left(\begin{array}{l}0 \\ 1 \\ c_{2}\end{array}\right), \boldsymbol{\alpha}_{3}=\left(\begin{array}{c}1 \\ -1 \\ c_{3}\end{array}\right), \boldsymbol{\alpha}_{4}=\left(\begin{array}{c}-1 \\ 1 \\ c_{4}\end{array}\right)$, 其中 $c_{1}, c_{2}, c_{3}, c_{4}$ 为任意常数, 则下列向量组线 性相关的为 ( )
$A.$ $\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}, \boldsymbol{\alpha}_{3}$. $B.$ $\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}, \boldsymbol{\alpha}_{4}$. $C.$ $\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{3}, \boldsymbol{\alpha}_{4}$. $D.$ $\boldsymbol{\alpha}_{2}, \boldsymbol{\alpha}_{3}, \boldsymbol{\alpha}_{4}$.
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答案:
C

解析:

解 $\quad\left|\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{3}, \boldsymbol{\alpha}_{4}\right|=\left|\begin{array}{ccc}0 & 1 & -1 \\ 0 & -1 & 1 \\ C_{1} & C_{3} & C_{4}\end{array}\right|=C_{1}\left|\begin{array}{rc}1 & -1 \\ -1 & 1\end{array}\right|=0$ 故 $\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{3}, \boldsymbol{\alpha}_{4}$ 线性相关. 故应选 C.

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