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How could I approach questions like the following?

If the first and third column of the matrix B are the same, will this be the case for the matrix AB as well ?

I'm not necessarily looking for a ready answer to this question. I just want to get an idea of how I should approach questions like this. Taking two random matrices and multiplying them wouldn't help alot, I think, because they can't prove anything. I would be only able to claim that for this case only the answer to the question is yes, but I'm looking for a way to get a certain answer for all cases. If I am to prove it wrong, a simple example of a case, where the answer is no, is enough. If such a case exists how am I to find hints about which exactly is this case?

Thank-you in advance.

PS. I'm not a native english speaker and everything I have done on linear algebra I have done it in my own language (Greek), so please forgive me if I use the wrong terminology. Just inform me of any mistakes I have done and I will repair them at once. If I haven't given enough or clear enough information, please don't hesitate to ask for clarifications.

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If $v$ is the first (and third) column of the matrix $B$, what will be the first column of $AB$? And the third? – Davide Giraudo Jan 22 '12 at 12:24
@DavideGiraudo : I guess it will be the product of the multiplication of the first row of a random A with the first/third column of B. The only thing I know about A is that it's length(number of columns) has to be the same as B's height(number of rows) so that I can multiply. – Eternal_Light Jan 22 '12 at 12:26
up vote 2 down vote accepted

Yes. They will be the same. You can think of the product $\mathbf{AB}$ as $\mathbf{A}$ independently acting on the columns of $\mathbf{B}$ and placing each product in the corresponding column of $\mathbf{AB}$. So a change to one column of $\mathbf{B}$ only affects that column. If we could denote the $i$th and $j$th columns of $\mathbf{B}$ as $\mathbf{b}_i$ and $\mathbf{b}_j$ then the corresponding columns of the product would likewise be $\mathbf{Ab}_i$ and $\mathbf{Ab}_j$. So if $\mathbf{b}_i = \mathbf{b}_j$ then they will be the same in product $\mathbf{AB}$. Work through the definition of matrix multiplication and you will see why this is the case. Pictorially you can view it as follows:

$$[\cdots\mathbf{Ab}_i \cdots \mathbf{Ab}_j \cdots] = \mathbf{A}[\cdots\mathbf{b}_i \cdots \mathbf{b}_j \cdots] $$

By the way the same could be said for the rows of $\mathbf{A}$ in the product $\mathbf{AB}$. Can you see why?

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I guess because B is independently acting on the rows of A? I understand it, I think. It looks like it is exactly the same thing but from A's point of view? – Eternal_Light Jan 22 '12 at 12:35

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