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 $A_1$ be an $m \times s$ matrix, $A_2$ be an $m \times (n-s)$ matrix, $B_1$ be an $s \times r$ matrix, and $B_2$ be an $(n-s) \times r$ matrix.
Prove that$$\begin{matrix} (A_1&A_2)\end{matrix} \begin{pmatrix} B_1\\B_2 \end{pmatrix} = A_1B_1+A_2B_2$$ 

Also let $A_{11}$ be an $k \times s$ matrix, $A_{12}$ be an $k \times (n-s)$ matrix, $A_{21}$ be an $(m-k) \times s$ matrix, $A_{22}$ be an $(m-k) \times (n-s)$ matrix, $B_{11}$ be an $s \times t$ matrix, $B_{12}$ be an $s \times (r-t)$ matrix, $B_{21}$ be an $(n-s) \times t$ matrix, and $B_{22}$ be an $(n-s) \times (r-t)$ matrix
Prove that
$$\begin{pmatrix} 
A_{11} & A_{12}\\ 
A_{21} & A_{22}  
\end{pmatrix}
\begin{pmatrix} 
B_{11} & B_{12}\\ 
B_{21} & B_{22}  
\end{pmatrix} = 
\begin{pmatrix} 
A_{11}B_{11}+A_{12}B_{21} & A_{11}B_{12} + A_{12}B_{22}\\ 
A_{21}B_{11} +A_{22}B_{21} & A_{22}B_{12}+A_{22}B_{22}  
\end{pmatrix}$$

I'm really with matrices and really need some guidance on how the above can be proven (can't even start the question). Thanks.
 A: 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}$$
