一、题目
已知,线性方程组 $A x=\alpha$ 有解, $\left(\begin{array}{l}A \\ B\end{array}\right) x=\left(\begin{array}{l}\alpha \\ \beta\end{array}\right)$ 无解,则下列结论中正确的是哪个?
A. $r(B, \beta)=r(B)+1$
B. $r\left(\begin{array}{ll}A & \alpha \\ B & \beta\end{array}\right)<r\left(\begin{array}{l}A \\ B\end{array}\right)+1$
C. $r\left[B^{\mathrm{\top}}(B, \beta)\right]>r\left(B^{\mathrm{\top}} B\right)$
D. $r\left[\left(A^{\mathrm{\top}}, B^{\mathrm{\top}}\right)\left(\begin{array}{ll}A & \alpha \\ B & \beta\end{array}\right)\right]=r\left[\left(A^{\mathrm{\top}}, B^{\mathrm{\top}}\right)\left(\begin{array}{l}A \\ B\end{array}\right)\right]$
难度评级:
继续阅读“做了这道题,你对分块矩阵性质的理解很可能将会更上一层楼”