3.单选题设|A|=|}（alpha_(1),alpha_(1)）（alpha_(1),alpha_(2)）（alpha_(2),alpha_(1)）（alpha_(2),alpha_(2)）

本题考查向量组线性相关性与行列式的关系，解题的关键在于理解向量内积构成的行列式与向量组线性相关性之间的联系，通过行列式的值是否为零来判断向量组的线性相关性。

设$\alpha_{1}=(x_1,y_1)^T$，$\alpha_{2}=(x_2,y_2)^T$，则$(\alpha_{1},\alpha_{1}) = x_1^2 + y_1^2$，$(\alpha_{1},\alpha_{2}) = x_1x_2 + y_1y_2$，$(\alpha_{2},\alpha_{1}) = x_2x_1 + y_2y_1$，$(\alpha_{2},\alpha_{2}) = x_2^2 + y_2^2$。

那么$\vert A\vert=\begin{vmatrix}(\alpha_{1},\alpha_{1})&(\alpha_{1},\alpha_{2})\\(\alpha_{2},\alpha_{1})&(\alpha_{2},\alpha_{2})\end{vmatrix}= (\alpha_{1},\alpha_{1})(\alpha_{2},\alpha_{2}) - (\alpha_{1},\alpha_{2})^2$。

根据柯西 - 施瓦茨不等式$(\alpha_{1},\alpha_{2})^2\leqslant (\alpha_{1},\alpha_{1})(\alpha_{2},\alpha_{2})$，当且仅当$\alpha_{1}$与$\alpha_{2}$线性相关时等号成立。

1. 充分性证明（若$\alpha_{1},\alpha_{2}$线性无关，则$\vert A\vert\neq0$）
   若$\alpha_{1},\alpha_{2}$线性无关，则$\alpha_{1}$与$\alpha_{2}$不成比例，此时$(\alpha_{1},\alpha_{2})^2\lt (\alpha_{1},\alpha_{1})(\alpha_{2},\alpha_{2})$，所以$\vert A\vert = (\alpha_{1},\alpha_{1})(\alpha_{2},\alpha_{2}) - (\alpha_{1},\alpha_{2})^2\gt 0$，即$\vert A\vert\neq0$。

2. 必要性证明（若$\vert A\vert\neq0$，则$\alpha_{1},\alpha_{2}$线性无关）
   若$\vert A\vert\neq0$，即$(\alpha_{1},\alpha_{1})(\alpha_{2},\alpha_{2}) - (\alpha_{1},\alpha_{2})^2\neq0$，也就是$(\alpha_{1},\alpha_{2})^2\neq (\alpha_{1},\alpha_{1})(\alpha_{2},\alpha_{2})$，根据柯西 - 施瓦茨不等式等号成立的条件可知，$\alpha_{1}$与$\alpha_{2}$不成比例，所以$\alpha_{1},\alpha_{2}$线性无关。

综上，$\vert A\vert\neq0$的充分必要条件是$\alpha_{1},\alpha_{2}$线性无关。