线性相关的向量组成的行列式一定等于零

题目 02

若 $\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}$, $\boldsymbol{\alpha}_{3}$ 线性无关, 若 $\boldsymbol{\alpha}_{1}+2 \boldsymbol{\alpha}_{2}+\boldsymbol{\alpha}_{3}$, $\boldsymbol{\alpha}_{1}+a \boldsymbol{\alpha}_{2}, 3 \boldsymbol{\alpha}_{2}+\boldsymbol{\alpha}_{3}$ 线性相关, 则 $a=?$

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解析 02

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

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

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

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

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