题目 07
已知三阶矩阵 $\boldsymbol{A}=\left(\boldsymbol{\alpha}, \boldsymbol{\gamma}_{1}, \boldsymbol{\gamma}_{2}\right)$, $\boldsymbol{B}=\left(\boldsymbol{\beta}, \boldsymbol{\gamma}_{1}, \boldsymbol{\gamma}_{2}\right)$, 其中 $\boldsymbol{\alpha}, \boldsymbol{\beta}, \boldsymbol{\gamma}_{1}, \boldsymbol{\gamma}_{2}$ 是三维列向量, 且 $|\boldsymbol{A}|=3$, $|\boldsymbol{B}|=4$, 则 $|5 \boldsymbol{A}-2 \boldsymbol{B}|=?$
解析 07
可以根据题意举个特例，例如，令：
$$
A=\left[\begin{array}{lll}3 & & \\ & 1 & 1\end{array}\right]
$$
$$
B=\left[\begin{array}{lll}4 & & \\ & & 1\end{array}\right]
$$
则：
$$
|5 A-2 B|=\left|\begin{array}{lll}7 & & \\ & 3 & \\ & & 3\end{array}\right|=63.
$$
或者直接算：
$$
5 A-2 B=
$$
$$
\left(5 \alpha, 5 \gamma_{1}, 5 \gamma_{2}\right)- \left(2 \beta, 2 \gamma_{1}, 2 \gamma_{2}\right)=
$$
$$
\left(5 \alpha -2 \beta, 3 \gamma_{1}, 3 \gamma_{2}\right)
$$
于是：
$$
|5 A-2 B|=\left|5 \alpha-2 \beta, 3 \gamma_{1}, 3 \gamma_{2}\right|=
$$
$$
9\left|5 \alpha-2 \beta, \gamma_{1}, \gamma_{2}\right|=
$$
$$
9\left(5\left|\alpha, \gamma_{1}, \gamma_{2}\right|-2\left|\beta, \gamma_{1}, \gamma_{2}\right|\right)=
$$
$$
9(5|A|-2|B|)=9(15-8)=63.
$$