设矩阵 $A, B$ 均为 $n$ 阶方阵, 若 $A x=0$ 与 $B x=0$ 同解, 则 ( ).
A. $\left(\begin{array}{ll}A & O \\ E & B\end{array}\right) x=0$ 仅有零解
B. $\left(\begin{array}{cc}A B & B \\ O & A\end{array}\right) x=0$ 仅有零解
C. $\left(\begin{array}{ll}A & B \\ O & B\end{array}\right) x=0$ 与 $\left(\begin{array}{ll}B & A \\ O & A\end{array}\right) x=0$ 同解
D. $\left(\begin{array}{cc}A B & B \\ O & A\end{array}\right) x=0$ 与 $\left(\begin{array}{cc}B A & A \\ O & B\end{array}\right) x=0$ 同解
设 $y=\left(\begin{array}{l}x_{1} \\ x_{2}\end{array}\right)$, 这里 $x_{i}(i=1,2)$ 是 $n$ 维列向量.
若 $\left(\begin{array}{ll}A & B \\ O & B\end{array}\right) y=0$ 与 $\left(\begin{array}{ll}B & A \\ O & A\end{array}\right) y=0$ 同解即 $\left(\begin{array}{ll}A & B \\ O & B\end{array}\right)\left(\begin{array}{l}x_{1} \\ x_{2}\end{array}\right)=0$ 与 $\left(\begin{array}{ll}B & A \\ O & A\end{array}\right)\left(\begin{array}{l}x_{1} \\ x_{2}\end{array}\right)=0$ 同
解. 由于 $A x=0$ 与 $B x=0$ 同解, 若 $A x=0 \quad x_{i}(i=1,2)$, 则 $B x_{i}=0(i=1,2)$, 反之亦然. 因
此 $\left(\begin{array}{ll}A & B \\ O & B\end{array}\right)\left(\begin{array}{l}x_{1} \\ x_{2}\end{array}\right)=0$ 等价于 $\left(\begin{array}{ll}B & A \\ O & A\end{array}\right)\left(\begin{array}{l}x_{1} \\ x_{2}\end{array}\right)=0$, 所以(C)选项符合题意