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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.

kevinbobbkoh
  • 357

1 Answers1

6

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}$$

Hypergeometricx
  • 22,657