$n$ 维行向量 $\boldsymbol{\alpha}=\left(\frac{1}{2}, 0, \ldots, 0, \frac{1}{2}\right), \mathbf{A}=\mathbf{E}-\boldsymbol{\alpha}^T \boldsymbol{\alpha}, \mathbf{B}=\mathbf{E}+2 \boldsymbol{\alpha}^T \boldsymbol{\alpha}$, 则 $\mathbf{A B}=$
$\text{A.}$ $\mathbf{0}$
$\text{B.}$ $\mathbf{E}$
$\text{C.}$ $-\mathbf{E}$;
$\text{D.}$ $5\mathbf{E}$;