已知 $n$ 阶矩阵 $A, B, C$ 满足 $A B C=0, E$ 为 $n$ 阶单位矩阵, 记矩阵 $\left(\begin{array}{cc}0 & A \\ B C & E\end{array}\right)$, $\left(\begin{array}{cc}A B & C \\ 0 & E\end{array}\right),\left(\begin{array}{cc}E & A B \\ A B & 0\end{array}\right)$ 的秩分别为 $\gamma_{1}, \gamma_{2}, \gamma_{3}$, 则()
(A) $\gamma_{1} \leq \gamma_{2} \leq \gamma_{3}$
(B) $\gamma_{1} \leq \gamma_{3} \leq \gamma_{2}$
(C) $\gamma_{3} \leq \gamma_{1} \leq \gamma_{2}$
(D) $\gamma_{2} \leq \gamma_{1} \leq \gamma_{3}$
因初等变换不改变矩阵的秩,
$r_{1}=r\left[\begin{array}{cc}0 & A \\ B C & E\end{array}\right]=r\left[\begin{array}{cc}-A B C & 0 \\ B C & E\end{array}\right]=r\left[\begin{array}{cc}0 & 0 \\ B C & E\end{array}\right]=n$,

$r_{2}=r\left[\begin{array}{cc}A B & C \\ 0 & E\end{array}\right]=r\left[\begin{array}{cc}A B & 0 \\ 0 & E\end{array}\right]=r(A B)+n$,

$r_{3}=r\left[\begin{array}{cc}E & A B \\ A B & 0\end{array}\right]=r\left[\begin{array}{cc}E & 0 \\ A B & -A B A B\end{array}\right]=r\left[\begin{array}{cc}E & 0 \\ 0 & -A B A B\end{array}\right]=r(A B A B)+n$,

故选(B).