# 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

## 一、题目

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. $r\left[\left(A^{\mathrm{\top}}, B^{\mathrm{\top}}\right)\left(\begin{array}{ll}A & \alpha \\ B & \beta\end{array}\right)\right]=r\left[\left(A^{\mathrm{\top}}, B^{\mathrm{\top}}\right)\left(\begin{array}{l}A \\ B\end{array}\right)\right]$

## 二、解析

### A 选项

$$\left(\begin{array}{l}A \\ B\end{array}\right) x=\left(\begin{array}{l}\alpha \\ \beta\end{array}\right) \Leftrightarrow \begin{cases} & A x = \alpha \\ & B x = \beta \end{cases}$$

$$\begin{cases} & A x = \alpha \\ & B x = \beta \end{cases}$$

（其中，$n$ 表示方程组中未知数的个数，或者系数矩阵的列数。）

### B 选项

$$r\left(\begin{array}{l}A \\ B\end{array}\right) < r\left(\begin{array}{ll}A & \alpha \\ B & \beta\end{array}\right)$$

$$r\left(\begin{array}{l}A \\ B\end{array}\right) + 1 = r\left(\begin{array}{ll}A & \alpha \\ B & \beta\end{array}\right)$$

$$r\left(\begin{array}{ll}A & \alpha \\ B & \beta\end{array}\right)<r\left(\begin{array}{l}A \\ B\end{array}\right)+1$$

### C 选项

$$r(A)=r\left(A^{\top} A\right)=r\left(A A^{\top}\right)=r\left(A^{\top}\right)$$

$$r(A B) \leqslant r(A) \quad r(A B) \leqslant r(B)$$

$$r[A, B] \geqslant r(A) \quad r[A, B] \geqslant r(B)$$

$$r\left[B^{\top}(B, \beta)\right]=r\left(B^{\top} B, B^{\top} \beta\right) \leqslant r\left(B^{\top} B\right)$$

$$r\left(B^{\top} B\right)=r(B)$$

$$r\left(B^{\top} B, B^{\top} \beta\right) \leqslant r(B)=r\left(B^{\top} B\right) \Rightarrow$$

$$r\left[B^{\top}(B, \beta)\right] \leqslant r\left(B^{\top} B\right)$$

### D 选项（第一种解法）

$$\left(\begin{array}{c} A \\ B \end{array}\right)^{\top} = (A^{\top}, B^{\top}) \quad (A, B)^{\top} = \left(\begin{array}{c} A^{\top} \\ B^{\top} \end{array}\right)$$

$$r\left(\begin{array}{c} A \\ B \end{array}\right)<r\left(\begin{array}{ll} A & \alpha \\ B & \beta \end{array}\right)$$

$$r\left(\begin{array}{l} A \\ B \end{array}\right) = r\left(A^{\top}, B^{\top}\right)<r\left(\begin{array}{ll} A & \alpha \\ B & \beta \end{array}\right)$$

$$r\left[\left(A^{\top}, B^{\top}\right)\left(\begin{array}{ll} A & \alpha \\ B & \beta \end{array}\right)\right]=r\left(A^{\top}, B^{\top}\right)$$

$$r\left[\left(A^{\top}, B^{\top}\right)\left(\begin{array}{l} A \\ B \end{array}\right)\right]=r\left[\left(\begin{array}{l} A \\ B \end{array}\right)^{\top}\left(\begin{array}{l} A \\ B \end{array}\right)\right]=r\left(\begin{array}{l} A \\ B \end{array}\right)^{\top}=$$

$$r\left(A^{\top}, B^{\top}\right)$$

$$r\left[\left(A^{\top}, B^{\top}\right)\left(\begin{array}{ll} A & \alpha \\ B & \beta \end{array}\right)\right]=r\left[\left(A^{\top}, B^{\top}\right)\left(\begin{array}{l} A \\ B \end{array}\right)\right]$$

### D 选项（第二种解法）

$$r\left[\left(A^{\top}, B^{\top}\right)\right]=r\left[\left(A^{\top}, B^{\top}\right)\left(\begin{array}{l} A \\ B \end{array}\right)\right]$$

$$r\left[\left(A^{\top}, B^{\top}\right)\left(\begin{array}{ll} A & \alpha \\ B & \beta \end{array}\right)\right] \textcolor{orange}{\leq} r\left[\left(A^{\top}, B^{\top}\right)\left(\begin{array}{l} A \\ B \end{array}\right)\right]$$

$$r\left[\left(A^{\top}, B^{\top}\right)\left(\begin{array}{ll} B & \alpha \\ B & \beta \end{array}\right)\right]=$$

$$r\left[\left(A^{\top}, B^{\top}\right)\left(\begin{array}{l}A \\ B\end{array}\right),\left(A^{\top}, B^{\top}\right)\left(\begin{array}{l}\alpha \\ \beta\end{array}\right)\right] \textcolor{orange}{\ge} r\left[\left(A^{\top}, B^{\top}\right)\left(\begin{array}{l}A \ B\end{array}\right)\right]$$

$$r\left[\left(A^{\top}, B^{\top}\right)\left(\begin{array}{ll}A & \alpha \\ B & \beta\end{array}\right)\right]=r\left[\left(A^{\top}, B^{\top}\right)\left(\begin{array}{l}A \\ B\end{array}\right)\right]$$