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
$ \text{A.} $ $\gamma_1 \leq \gamma_2 \leq \gamma_3$ $ \text{B.} $ $\gamma_1 \leq \gamma_3 \leq \gamma_2$ $ \text{C.} $ $\gamma_3 \leq \gamma_1 \leq \gamma_2$ $ \text{D.} $ $\gamma_2 \leq \gamma_1 \leq \gamma_3$
【答案】 B

【解析】 因初等变换不改变矩阵的秩, $$
\begin{aligned}
& r_1=r\left[\begin{array}{cc}
0 & A \\
B C & E
\end{array}\right]=r\left[\begin{array}{cc}
-A B C & 0 \\
B C & E
\end{array}\right]=r\left[\begin{array}{cc}
0 & 0 \\
B C & E
\end{array}\right]=n, \\
& r_2=r\left[\begin{array}{cc}
A B & C \\
0 & E
\end{array}\right]=r\left[\begin{array}{cc}
A B & 0 \\
0 & E
\end{array}\right]=r(A B)+n, \\
& r_3=r\left[\begin{array}{cc}
E & A B \\
A B & 0
\end{array}\right]=r\left[\begin{array}{cc}
E & 0 \\
A B & -A B A B
\end{array}\right]=r\left[\begin{array}{cc}
E & 0 \\
0 & -A B A B
\end{array}\right]=r(A B A B)+n,
\end{aligned}
$$
故选(B).
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