设 $A, B$ 为 $n$ 阶方阵，$I$ 为 $n$ 阶单位矩阵，计算
$$
\left(\begin{array}{cc}
I & I \\
O & I
\end{array}\right)\left(\begin{array}{cc}
A & B \\
B & A
\end{array}\right)\left(\begin{array}{cc}
I & -I \\
O & I
\end{array}\right)
$$
并由此证明 $\left|\begin{array}{ll}A & B \\ B & A\end{array}\right|=|A+B| \cdot|A-B|$
$\text{A.}$
$\text{B.}$
$\text{C.}$
$\text{D.}$
$\text{E.}$
$\text{F.}$