Prove that an augmented matrix \begin{matrix} (A_1&A_2)\end{matrix} \begin{pmatrix} B_1\\B_2 \end{pmatrix} = A_1B_1+A_2B_2
Let A_1=P, A_2=Q, B_1=H, B_2=K. Then we have \scriptsize\begin{align} &\;\;\;\;(A_1\quad A_2)\left(B_1\atop B_2\right)\\\\ &=(P\quad Q)\;\;\;\left(H\atop K\right)\\\\ &=\left(\begin{array}{rrr:rrr} p_{1,1}&p_{1,2}&\cdots \;p_{1,s}&q_{1,1}&q_{1,2}&\cdots \;q_{1,n-s} \\ p_{2,1}&p_{2,2}&\cdots \;p_{2,s}&q_{2,1}&q_{2,2}&\cdots \;q_{2,n-s} \\ &\vdots&&&\vdots\\ p_{m,1}&p_{m,2}&\cdots\; p_{m,s}&q_{m,1}&q_{m,2}&\cdots \;q_{m,n-s} \end{array}\right) \left(\begin{array}{rrr} h_{1,1}&h_{1,2}&\cdots \; h_{1,r}\\ h_{2,1}&h_{2,2}&\cdots \; h_{2,r}\\ &\vdots \\ h_{s,1}&h_{s,2}&\cdots \; h_{s,r}\\ \hdashline k_{1,1}&k_{1,2}&\cdots \; k_{1,r}\\ k_{2,1}&k_{2,2}&\cdots \; k_{2,r}\\ &\vdots \\ k_{n-s,1}&k_{n-s,2}&\cdots \; k_{n-s,r}\\ \end{array}\right)\\\\ &=\underbrace{\left.\left(\begin{array} .\boxed{\sum_{i=1}^s p_{1,i}h_{i,1}+\sum_{j=1}^{n-s}q_{1,j}k_{j,1}} &\boxed{\sum_{i=1}^s p_{1,i}h_{i,2}+\sum_{j=1}^{n-s}q_{1,j}k_{j,2}} &\cdots &\boxed{\sum_{i=1}^s p_{1,i}h_{i,r}+\sum_{j=1}^{n-s}q_{1,j}k_{j,r}}\\ \boxed{\sum_{i=1}^s p_{2,i}h_{i,1}+\sum_{j=1}^{n-s}q_{2,j}k_{j,1}} &\boxed{\sum_{i=1}^s p_{2,i}h_{i,2}+\sum_{j=1}^{n-s}q_{2,j}k_{j,2}} &\cdots &\boxed{\sum_{i=1}^s p_{2,i}h_{i,r}+\sum_{j=1}^{n-s}q_{2,j}k_{j,r}}\\ \qquad\qquad\vdots&\qquad\qquad\vdots&\cdots &\qquad\qquad\vdots\\ \boxed{\sum_{i=1}^s p_{m,i}h_{i,1}+\sum_{j=1}^{n-s}q_{m,j}k_{j,1}} &\boxed{\sum_{i=1}^s p_{m,i}h_{i,2}+\sum_{j=1}^{n-s}q_{m,j}k_{j,2}} &\cdots &\boxed{\sum_{i=1}^s p_{m,i}h_{i,r}+\sum_{j=1}^{n-s}q_{m,j}k_{j,r}}\\ \end{array}\right)\;\right\}}_{r\text{ columns}} \,m\text{ rows}\\\\ &=\left(\begin{array} .\boxed{\sum_{i=1}^s p_{1,i}h_{i,1}} &\boxed{\sum_{i=1}^s p_{1,i}h_{i,2}} &\cdots &\boxed{\sum_{i=1}^s p_{1,i}h_{i,r}}\\ \boxed{\sum_{i=1}^s p_{2,i}h_{i,1}} &\boxed{\sum_{i=1}^s p_{2,i}h_{i,2}} &\cdots &\boxed{\sum_{i=1}^s p_{2,i}h_{i,r}}\\ \;\quad\vdots&\;\quad\vdots&\cdots &\;\quad\vdots\\ \boxed{\sum_{i=1}^s p_{m,i}h_{i,1}} &\boxed{\sum_{i=1}^s p_{m,i}h_{i,2}} &\cdots &\boxed{\sum_{i=1}^s p_{m,i}h_{i,r}}\\ \end{array}\right)\; \,\longleftarrow PH\\\\ &\; +\left(\begin{array} .\boxed{\sum_{j=1}^{n-s}q_{1,j}k_{j,1}} &\boxed{\sum_{j=1}^{n-s}q_{1,j}k_{j,2}} &\cdots &\boxed{\sum_{j=1}^{n-s}q_{1,j}k_{j,r}}\\ \boxed{\sum_{j=1}^{n-s}q_{2,j}k_{j,1}} &\boxed{\sum_{j=1}^{n-s}q_{2,j}k_{j,2}} &\cdots &\boxed{\sum_{j=1}^{n-s}q_{2,j}k_{j,r}}\\ \;\quad\vdots&\;\quad\vdots&\cdots &\;\quad\vdots\\ \boxed{\sum_{j=1}^{n-s}q_{m,j}k_{j,1}} &\boxed{\sum_{j=1}^{n-s}q_{m,j}k_{j,2}} &\cdots &\boxed{\sum_{j=1}^{n-s}q_{m,j}k_{j,r}}\\ \end{array}\right)\; \,\longleftarrow QK\\\\ &=PH+QK\\\\ &=A_1B_1+A_2B_2 \end{align}
