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My textbook claims:

claim

I don't understand how D can be true. Say I have a 2x4 matrix and I multiply it by the 4x2 zero matrix. I get a 2x2 zero matrix. Now assume I multiplied in the other order, I multiply the 4x2 zero matrix against the 2x4 matrix. I get the 4x4 zero matrix. They are both zero matrices, but they have different dimensions! Surely they can't be considered equal in this case?

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    $\begingroup$ You are correct $\endgroup$ – David Peterson May 24 '17 at 1:26
  • $\begingroup$ @DavidP answer with an answer instead of a comment and I will mark correct $\endgroup$ – Joseph Garvin May 24 '17 at 1:27
  • $\begingroup$ your textbook indicates that the matrices must be of an appropriate size. Sorry, you matrices are of an inappropriate size ;) But, you are right, as operations could indeed be carried out in the example that you give :) $\endgroup$ – Mirko May 24 '17 at 3:21
  • $\begingroup$ @littleo Actually you can reverse the order of multiplication even in cases where they aren't square. $p\times q$ and $q\times p$ can be multiplied in either order. $\endgroup$ – Matt Samuel May 24 '17 at 22:12
  • $\begingroup$ @MattSamuel Oh yeah, thanks, I'm going to delete the comment. $\endgroup$ – littleO May 25 '17 at 0:31
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The textbook says that we're working with matrices of appropriate sizes. You're 100% correct if $\mathbf{A}$ is not a square matrix, but if $\mathbf{A}$ is a square matrix then multiplication by the zero matrix is indeed commutative.

However, that doesn't mean your textbook is off the hook. They're explaining this property really clumsily; it would be great if they specified which matrices they're talking about, rather than just assuming the reader knows what the "appropriate size" is.

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Wellll, I guess the textbook could be considered correct if it is implicitly saying that your example consisted of matrices of 'inappropriate' size. One could make the argument that the equals sign doesn't really make sense for matrices of different sizes, so the whole statement is vacuous. But your intuition is correct, and this is an example of slightly sloppy wording by the author.

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