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Let $A_1,A_2,R_1,R_2$ be $m\times n$ matrices. Is the following true always?

$R_1(A_1+A_2)R_2 = R_1A_1R_2 + R_1A_2R_2$

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    Minor typo about sizes.2011-10-15

2 Answers 2

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I presume you mean $R_1$ is $m\times m$, the matrices $A_1$ and $A_2$ are $m\times n$ and $R_2$ is $n\times n$ (otherwise the matrices do not have conforming sizes for matrix multiplication to be defined). Then the answer to your question is affirmative. This is because matrix multiplication itself is assocative and it is distributive over matrix addition.

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Yes. It follows from associativity and distributivity: $A(X+Y)B=(AX+AY)B= AXB+AYB,\text{ or}$ $A(X+Y)B=A(XB+YB)=AXB+AYB.\;\;\;\;\text{ }$

For general not-necessarily square matrices, distributivity should be obvious through distributing through the implict summation, and associativity can likewise be checked by writing out multiplication with indices and interchanging the order of summation: $[(AB)C]_{i\ell}=\sum_{k=1}^m\left(\sum_{j=1}^na_{ij}b_{jk}\right)c_{k\ell}=\sum_{j=1}^na_{ij}\left(\sum_{k=1}^mb_{jk}c_{k\ell}\right)=[A(BC)]_{i\ell}. $