行列式和矩阵的计算规则有什么区别？做了这道题就明白了！

一、题目

已知，四阶方阵 $\boldsymbol{A}$ $=$ $\left[\boldsymbol{\alpha}, \boldsymbol{\gamma}_{2}, \boldsymbol{\gamma}_{3}, \boldsymbol{\gamma}_{4}\right]$, $\boldsymbol{B}$ $=$ $\left[\boldsymbol{\beta}, \boldsymbol{\gamma}_{2}, \boldsymbol{\gamma}_{3}, \boldsymbol{\gamma}_{4}\right]$, 其中 $\boldsymbol{\alpha}$, $\boldsymbol{\beta}$, $\boldsymbol{\gamma}_{2}$, $\boldsymbol{\gamma}_{3}$, $\boldsymbol{\gamma}_{4}$ 均为四维列向量，且 $|\boldsymbol{A}|=5$, $|\boldsymbol{B}|=-\frac{1}{2}$, 则 $|\boldsymbol{A}+2 \boldsymbol{B}| = ?$

难度评级：

二、解析

$$
2 B=\left[2 \beta, 2 \gamma_{2}, 2 \gamma_{3}, 2 \gamma_{4}\right] \Rightarrow
$$

$$
A + 2 B=\left[\alpha+2 \beta, 3 \gamma_{2}, 3 \gamma_{3}, 3 \gamma_{4}\right] \Rightarrow
$$

$$
|A+2 B|=3^{3}\left|\alpha+2 \beta, \gamma_{2}, \gamma_{3}, \gamma_{4}\right| =
$$

$$
3^{3}\left|\alpha, \gamma_{2}, \gamma_{3}, \gamma_{4}\right|+3^{3}\left|2 \beta, \gamma_{2}, \gamma_{3}, \gamma_{4}\right| =
$$

$$
27|A|+3^{3} \times 2\left|\beta, \gamma_{2}, \gamma_{3}, \gamma_{4}\right| =
$$

$$
27 \times 5+27 \times 2 \times|B| =
$$

$$
27 \times 5+27 \times 2 \times\left(-\frac{1}{2}\right) =
$$

$$
27 \times 4=108.
$$

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