设 $\alpha_{1}, \alpha_{2}, \alpha_{3}$ 是 3 维向量空间 $\mathbf{R}^{3}$ 的一组基, 则由基 $\alpha_{1}, \frac{1}{2} \alpha_{2}, \frac{1}{3} \alpha_{3}$ 到基 $\alpha_{1} + \alpha_{2}, \alpha_{2} + \alpha_{3}, \alpha_{3} + \alpha_{1}$ 的过渡矩阵为 ( ) (A) $\left( \begin{array}{lll}1 & 0 & 1\\ 2 & 2 & 0\\ 0 & 3 & 3 \end{array} \right)$ (B) $\left( \begin{array}{lll}1 & 2 & 0\\ 0 & 2 & 3\\ 1 & 0 & 3 \end{array} \right)$ $$ \mathrm {(C)} \left( \begin{array}{c c c} \frac {1}{2} & \frac {1}{4} & - \frac {1}{6} \\ - \frac {1}{2} & \frac {1}{4} & \frac {1}{6} \\ \frac {1}{2} & - \frac {1}{4} & \frac {1}{6} \end{array} \right). $$ $$ \text {(D)} \left( \begin{array}{c c c} \frac {1}{2} & - \frac {1}{2} & \frac {1}{2} \\ \frac {1}{4} & \frac {1}{4} & - \frac {1}{4} \\ - \frac {1}{6} & \frac {1}{6} & \frac {1}{6} \end{array} \right). $$