线性相关的向量组成的行列式一定等于零

题目 02

若 ${\mathit{\alpha }}_1,{\mathit{\alpha }}_2$, ${\mathit{\alpha }}_3$ 线性无关, 若 ${\mathit{\alpha }}_1+2{\mathit{\alpha }}_2+{\mathit{\alpha }}_3$, ${\mathit{\alpha }}_1+a{\mathit{\alpha }}_2,3{\mathit{\alpha }}_2+{\mathit{\alpha }}_3$ 线性相关, 则 $a=?$

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解析 02

$$({\alpha }_1,{\alpha }_2,{\alpha }_3)\left[\begin{array}{lll}1& 1& 0\\ 2& a& 3\\ 1& 0& 1\end{array}\right]\Rightarrow$$

$$\left|\begin{array}{lll}1& 1& 0\\ 2& a& 3\\ 1& 0& 1\end{array}\right|=0\Rightarrow$$

$$\left|\begin{array}{ccc}1& 1& 0\\ 0& a-2& 3\\ 1& 0& 1\end{array}\right|=0\Rightarrow$$

$$a-2+3=0\Rightarrow a=-1$$

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