# Matrix Multiplication - Why Rows $\cdot$ Columns = Columns?

I'm nearing the end of my first year of Calculus and am pretty confident in the parts of it I've learned, yet I still don't have a good understanding of matrices, which seem like they should be easier to understand and work with. They were never formally taught in any of the math courses my school has given me and were only ever briefly mentioned.

I don't fully understand what they represent or the logic behind how they are multiplied. Why are the left matrix's rows multiplied by the second matrix's columns to form columns in the resulting matrix? Why not columns by rows to form rows? Was it an arbitrary decision that was made or does it fit what concept of the matrix?

I could memorize how to perform operations on matrices and how to use them to solve certain problems, but what I'm trying to do is understand them conceptually.

Specifically, I was reading a page about the matrix representation of 2d transformations as used by CSS3 and found it all very confusing.

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You might find this question interesting: math.stackexchange.com/questions/31725/… –  Adam Saltz Apr 29 '13 at 5:51
@AdamSaltz, hmm is there an area of mathematics I should understand before attempting to understand matrices? What exactly did that answerer mean by linear function? Just a function with variables all to the first power? –  mowwwalker Apr 29 '13 at 5:53
This is all the subject of linear algebra, so that's a good place to start. I didn't mean to link you to that specific answer -- other answers might be more relevant to you. Your question is pretty common: math.stackexchange.com/questions/271927/… –  Adam Saltz Apr 29 '13 at 5:56
@AdamSaltz, In that case, do you know of any good online resources for learning linear algebra? –  mowwwalker Apr 29 '13 at 6:03
I think the Wikipedia page on matrices is a pretty good place to start. I don't know if I'd recommend learning linear algebra first. Linear algebra involves abstractions/generalizations of matrices, but, given your current understanding and interests, this is probably the wrong direction, for now. –  bubba Apr 29 '13 at 6:22

Suppose you have a crate with $100$ apples and $200$ oranges, and another crate with $150$ apples and $80$ oranges. You can represent this situation with the matrix $$\pmatrix{100&200\cr150&80\cr}$$ Now suppose apples weigh $4$ ounces each and cost $50$ cents each, while oranges weigh $5$ ounces each and cost $60$ cents each. You can represent this information by the matrix $$\pmatrix{4&50\cr5&60\cr}$$ The total weight and total value of the fruit in the first crate are given by $(100)(4)+(200)(5)=1400$ ounces and $(100)(50)+(200)(60)=17000$ cents, respectively; for the second crate, $(150)(4)+(80)(5)=1000$ ounces and $(150)(50)+(80)(60)=12300$ cents.
Well, we've just done matrix multiplication: $$\pmatrix{100&200\cr150&80\cr}\pmatrix{4&50\cr5&60\cr}=\pmatrix{1400&17000\cr1000&12300\cr}$$