设 $A, P$ 均为 3 阶矩阵， $P^T$ 为 $P$ 的转置矩阵，且 $P^T A P=\left(\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2\end{array}\right)$, 若
$$
P=\left(\alpha_1, \alpha_2, \alpha_3\right), Q=\left(\alpha_1+\alpha_2, \alpha_2, \alpha_3\right),
$$
则 $Q^T A Q$ 为
A. $\left(\begin{array}{lll}2 & 1 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 2\end{array}\right)$
B. $\left(\begin{array}{lll}1 & 1 & 0 \\ 1 & 2 & 0 \\ 0 & 0 & 2\end{array}\right)$
C. $\left(\begin{array}{lll}2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2\end{array}\right)$
D. $\left(\begin{array}{lll}1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2\end{array}\right)$