已知向量 alpha_1, alpha_2, alpha_3, alpha_4, alpha_5，且有 r(alpha_1, alpha_2, alpha_3, alpha_4) = 3，r(alpha_1, alpha_2, alpha_3, alpha_5) = 4，则 r(alpha_1, alpha_2, alpha_3, alpha_4, alpha_5) = ( )A. 4B. 1C. 3D. 2

本题考查向量组的秩的性质。解题的关键思路是根据已知向量组的秩，分析向量之间的线性关系，进而确定所求向量组的秩。

步骤一：分析$r(\alpha_1, \alpha_2, \alpha_3, \alpha_4) = 3$

向量组$\alpha_1, \alpha_2, \alpha_3, \alpha_4$的秩为$3$，这意味着向量组$\alpha_1, \alpha_2, \alpha_3, \alpha_4$中极大线性无关组所含向量的个数为$3$。不妨设$\alpha_1, \alpha_2, \alpha_3$是$\alpha_1, \alpha_2, \alpha_3, \alpha_4$的一个极大线性无关组，那么$\alpha_4$可由$\alpha_1, \alpha_2, \alpha_3$线性表示，即存在一组实数$k_1,k_2,k_3$，使得$\alpha_4 = k_1\alpha_1 + k_2\alpha_2 + k_3\alpha_3$。

步骤二：分析$r(\alpha_1, \alpha_2, \alpha_3, \alpha_5) = 4$

向量组$\alpha_1, \alpha_2, \alpha_3, \alpha_5$的秩为$4$，这表明向量组$\alpha_1, \alpha_2, \alpha_3, \alpha_5$线性无关。

步骤三：确定$r(\alpha_1, \alpha_2, \alpha_3, \alpha_4, \alpha_5)$

由于$\alpha_4$可由$\alpha_1, \alpha_2, \alpha_3$线性表示，所以向量组$\alpha_1, \alpha_2, \alpha_3, \alpha_4, \alpha_5$与向量组$\alpha_1, \alpha_2, \alpha_3, \alpha_5$等价。根据等价向量组的秩相等这一性质，可知$r(\alpha_1, \alpha_2, \alpha_3, \alpha_4, \alpha_5)=r(\alpha_1, \alpha_2, \alpha_3, \alpha_5)=4$。