这里的矩阵 B 可以由矩阵 A 变过来，你看出来了吗？

一、题目

已知，$\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}, \boldsymbol{\alpha}_{3}$ 为三维列向量,矩阵 $\boldsymbol{A}=\left[\boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2}, \boldsymbol{\alpha}_{3}\right], \boldsymbol{B}=\left[\boldsymbol{\alpha}_{3}, 2 \boldsymbol{\alpha}_{1}+\boldsymbol{\alpha}_{2}, 3 \boldsymbol{\alpha}_{2}\right]$, 若行列式 $|\boldsymbol{A}|=2$, 则行列式 $|\boldsymbol{B}|=?$

难度评级：

二、解析

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

$$
|B|=\left|\begin{array}{lll}0 & 0 & 1 \\ 2 & 1 & 0 \\ 0 & 3 & 0\end{array}\right| \cdot|A|=6 \times 2=12
$$

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