下证
\begin{equation*}
\begin{pmatrix}\alpha_{i_1}\\0\end{pmatrix},\dots,\begin{pmatrix}\alpha_{i_r}\\0\end{pmatrix},\begin{pmatrix}0\\\beta_{j_1}\end{pmatrix},\dots,\begin{pmatrix}0\\\beta_{j_p}\end{pmatrix}
\end{equation*}
是\(\begin{pmatrix} A & 0\\ 0 & B \end{pmatrix}\)列向量组\(\begin{pmatrix}\alpha_1\\0\end{pmatrix},\dots,\begin{pmatrix}\alpha_s\\0\end{pmatrix},\begin{pmatrix}0\\\beta_1\end{pmatrix},\dots,\begin{pmatrix}0\\\beta_t\end{pmatrix}\)的一个极大无关组。事实上，若
\begin{equation*}
a_1\begin{pmatrix}\alpha_{i_1}\\0\end{pmatrix} + \cdots + a_r\begin{pmatrix}\alpha_{i_r}\\0\end{pmatrix} + b_1\begin{pmatrix}0\\\beta_{j_1}\end{pmatrix} + \cdots + b_p\begin{pmatrix}0\\\beta_{j_p}\end{pmatrix}=0,
\end{equation*}
即 \(\begin{pmatrix}a_1\alpha_{i_1} + \cdots + a_r\alpha_{i_r}\\b_1\beta_{j_1} + \cdots + b_p\beta_{j_p}\end{pmatrix}=0\)，则
\begin{equation*}
a_1\alpha_{i_1} + \cdots + a_r\alpha_{i_r}=0\ \text{且}\ b_1\beta_{j_1} + \cdots + b_p\beta_{j_p}=0.
\end{equation*}
由向量组\(\alpha_{i_1},\dots,\alpha_{i_r}\)及\(\beta_{j_1},\dots,\beta_{j_p}\)线性无关知
\begin{equation*}
a_1=\cdots=a_r=b_1=\cdots=b_p=0,
\end{equation*}
因此向量组\(\begin{pmatrix}\alpha_{i_1}\\0\end{pmatrix},\dots,\begin{pmatrix}\alpha_{i_r}\\0\end{pmatrix},\begin{pmatrix}0\\\beta_{j_1}\end{pmatrix},\dots,\begin{pmatrix}0\\\beta_{j_p}\end{pmatrix}\)线性无关。对任意\(1\leq k\leq s\)，由\(\alpha_{i_1},\dots,\alpha_{i_r}\)是\(A\)的列向量组\(\alpha_1,\dots,\alpha_s\)的一个极大无关组知存在\(c_{1k},\dots,c_{rk}\in\mathbb{F}\)，使得
\begin{equation*}
\alpha_k=c_{1k}\alpha_{i_1} + \cdots + c_{rk}\alpha_{i_r},
\end{equation*}
于是
\begin{equation*}
\begin{pmatrix} \alpha_k \\ 0 \end{pmatrix}=c_{1k}\begin{pmatrix} \alpha_{i_1} \\ 0 \end{pmatrix}+ \cdots + c_{rk}\begin{pmatrix} \alpha_{i_r} \\ 0\end{pmatrix}+0\begin{pmatrix}0\\\beta_{j_1}\end{pmatrix} + \cdots + 0\begin{pmatrix}0\\\beta_{j_p}\end{pmatrix}.
\end{equation*}
同理可证对任意\(1\leq l\leq t\)，向量\(\begin{pmatrix}0\\\beta_l\end{pmatrix}\)可由向量组\(\begin{pmatrix}\alpha_{i_1}\\0\end{pmatrix},\dots,\begin{pmatrix}\alpha_{i_r}\\0\end{pmatrix},\begin{pmatrix}0\\\beta_{j_1}\end{pmatrix},\dots,\begin{pmatrix}0\\\beta_{j_p}\end{pmatrix}\)线性表出。因此\(\begin{pmatrix}\alpha_{i_1}\\0\end{pmatrix},\dots,\begin{pmatrix}\alpha_{i_r}\\0\end{pmatrix},\begin{pmatrix}0\\\beta_{j_1}\end{pmatrix},\dots,\begin{pmatrix}0\\\beta_{j_p}\end{pmatrix}\)是\(\begin{pmatrix} A & 0\\ 0 & B \end{pmatrix}\)列向量组的一个极大无关组，从而
\begin{equation*}
r\left(\begin{array}{cc}
A&0\\0&B
\end{array}\right)=r+p=r(A)+r(B).
\end{equation*}
