什么样的是充分条件？什么样的是必要条件？

一、题目

$a = 1$ 是向量组 $α_{1} = (1, 1, a)^{⊤}, α_{2} = (1, a, 1)^{⊤}, α_{3} = (a, 1, 1)^{⊤}, α_{4} = (- 2, - 2, a +$ $6)^{⊤}$ 的秩为 2 的充要条件吗？

难度评级：

$(α_{1}, α_{2}, α_{3}, α_{4}) \Rightarrow$

$[\begin{array}{llll} 1 & 1 & a & - 2 \\ 1 & a & 1 & - 2 \\ a & 1 & 1 & a + 6 \end{array}] \Rightarrow$

$[\begin{array}{cccc} 1 & 1 & a & - 2 \\ 0 & a - 1 & 1 - a & 0 \\ 0 & 1 - a & 1 - a^{2} & 3 a + 6 \end{array}] \Rightarrow$

$[\begin{array}{cccc} 1 & 1 & a & - 2 \\ 0 & a - 1 & 1 - a & 0 \\ 0 & 0 & 2 - a - a^{2} & 3 a + 6 \end{array}]$

$a = 1 \Rightarrow (\begin{array}{llll} 0 & a - 1 & 1 - a & 0 \end{array}) = (\begin{array}{llll} 0 & 0 & 0 & 0 \end{array})$

$a = - 2 \Rightarrow (\begin{array}{llll} 0 & 0 & 2 - a - a^{2} & 3 a + 6 \end{array}) = (\begin{array}{lll} 0 & 0 & 0 & 0 \end{array})$

所以，根据“前充分后必要，小充分大必要”的规则， $a = 1$ 是充分非必要条件。