设 $\alpha$ 为 3 维列向量， $\alpha^T$ 是 $\alpha$ 的转置，若
$$
\alpha \alpha^T=\left(\begin{array}{ccc}
1 & -1 & 1 \\
-1 & 1 & -1 \\
1 & -1 & 1
\end{array}\right) \text {. }
$$
则 $\boldsymbol{\alpha}^T \boldsymbol{\alpha}=$
$\text{A.}$
$\text{B.}$
$\text{C.}$
$\text{D.}$