题目
7.设alpha_(1),alpha_(2),alpha_(3),beta_(1),beta_(2)均为4维列向量，矩阵A=(alpha_(1),alpha_(2),alpha_(3),beta_(1))，B=(alpha_(1),alpha_(2),alpha_(3),beta_(2))，且|A|=3，|B|=-2，则|A+B|=____.
7.设$\alpha_{1},\alpha_{2},\alpha_{3},\beta_{1},\beta_{2}$均为4维列向量，矩阵$A=(\alpha_{1},\alpha_{2},\alpha_{3},\beta_{1})$，$B=(\alpha_{1},\alpha_{2},\alpha_{3},\beta_{2})$，且$|A|=3$，$|B|=-2$，则$|A+B|=$____.
题目解答
答案
将矩阵 $ A + B $ 表示为 $ (2\alpha_1, 2\alpha_2, 2\alpha_3, \beta_1 + \beta_2) $。利用行列式性质，提取前三个列向量的公因子2，得：
\[ |A + B| = 8 \cdot |\alpha_1, \alpha_2, \alpha_3, \beta_1 + \beta_2| \]
再由行列式对列向量的可加性，有：
\[ |\alpha_1, \alpha_2, \alpha_3, \beta_1 + \beta_2| = |A| + |B| = 3 - 2 = 1 \]
因此，$ |A + B| = 8 \cdot 1 = 8 $。
答案：$\boxed{8}$