线性无关的向量经运算之后变相关，则背后隐藏的矩阵一定线性相关

一、题目

已知 ${\mathit{\alpha }}_1,{\mathit{\alpha }}_2,{\mathit{\alpha }}_3$ 线性无关，若 ${\mathit{\alpha }}_1+{\mathit{\alpha }}_2,a{\mathit{\alpha }}_2-{\mathit{\alpha }}_3,{\mathit{\alpha }}_1-{\mathit{\alpha }}_2+{\mathit{\alpha }}_3$ 线性相关，则 $a=?$

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二、解析

$$({\alpha }_1+{\alpha }_2,a{\alpha }_2-{\alpha }_3,{\alpha }_1-{\alpha }_2+{\alpha }_3)=$$

$$({\alpha }_1,{\alpha }_2,{\alpha }_3)\left[\begin{array}{ccc}1& 0& 1\\ 1& a& -1\\ 0& -1& 1\end{array}\right]\Rightarrow$$

$$\left|\begin{array}{ccc}1& 0& 1\\ 1& a& -1\\ 0& -1& 1\end{array}\right|=0\Rightarrow \left|\begin{array}{ccc}1& 0& 1\\ 2& a& 0\\ 0& -1& 1\end{array}\right|=0\Rightarrow$$

$$a-2=0\Rightarrow a=2.$$

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