你会拆分这种行列式吗？

一、题目

已知 $\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}, \boldsymbol{\alpha}_{3}$ 是三维线性无关列向量，请问：

$\left|\boldsymbol{\alpha}_{1}+\boldsymbol{\alpha}_{2}, \boldsymbol{\alpha}_{2}+\boldsymbol{\alpha}_{3}, \boldsymbol{\alpha}_{3}+\boldsymbol{\alpha}_{1}\right| \neq 0$ 一定成立吗？

难度评级：

二、解析

$$
\left(\alpha_{1}, \alpha_{2}, \alpha_{3}\right)\left(\begin{array}{ccc}1 & 0 & 1 \\ 1 & 1 & 0 \\ 0 & 1 & 1\end{array}\right) = (\alpha_{1}+\alpha_{2}, \alpha_{2}+\alpha_{3}, \alpha_{3}+\alpha_{1}) \Rightarrow
$$

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

$$
\left|\alpha_{1}, \alpha_{2}, \alpha_{3}\right|\left|\begin{array}{lll}1 & 0 & 1 \\ 1 & 1 & 0 \\ 0 & 1 & 1\end{array}\right| \neq 0 \Rightarrow
$$

$$
\left|\alpha_{1}+\alpha_{2}, \alpha_{2}+\alpha_{3}, \alpha_{3}+\alpha_{1}\right| \neq 0
$$

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