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. 
 A: 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: $$\pmatrix{100&200\cr150&80\cr}\pmatrix{4&50\cr5&60\cr}=\pmatrix{1400&17000\cr1000&12300\cr}$$
So we see that matrix multiplication has been defined in such a way as to meet the needs of the produce industry, and everyone else has fallen into line with this important sector of the economy. 
A: I'm not entirely sure if this is the best answer for this question, because the $\text{Row}\cdot\text{Column}$ method has many important implications and is unavoidable if you carry on for more advanced uses of it, and those ideas might be a better way of dealing with the question.
But what happens if matrix multiplication was defined a different way? That is a good point to start for feeling comfortable with any topic.
Arthur Cayley wrote a book in 1857 called “A Memoir on the Theory of Matrices” which is believed to have the earliest printed description of matrix multiplication. 
And he uses an unconventional matrix multiplication method which might look a lot more intuitive for a beginner. He starts with stating that a matrix is used to describe a set of equations like:
$ax+by+cz = X$
$a'x+b'y+c'z = Y$
$a''x+b''y+c''z=Z$
And then describes that matrix as such:
$\begin{pmatrix} a&b&c\\a'&b'&c'\\a''&b''&c'' \end{pmatrix} \begin{pmatrix} x&y&z \end{pmatrix} = \begin{pmatrix} X&Y&Z \end{pmatrix} = \begin{pmatrix} (a,b,c)(x,y,z)&(a',b',c')(x,y,z)&(a'',b'',c'')(x,y,z) \end{pmatrix}$
It seems more understandable. You take rows and multiply it with corresponding row elements (and then add them up). In fact, that’s similar to how we define addition.
But then Cayley introduces another vector 
$\begin{pmatrix} \xi & \eta & \zeta \end{pmatrix}$ and then he states that,
$\begin{pmatrix} x&y&z \end{pmatrix} = \begin{pmatrix} \alpha & \beta & \gamma \\ \alpha ' & \beta ' & \gamma ' \\ \alpha '' & \beta '' & \gamma '' \end{pmatrix} \begin{pmatrix} \xi & \eta & \zeta \end{pmatrix}$
So the original $\begin{pmatrix} X&Y&Z \end{pmatrix}$ can also be seen as,
$\begin{pmatrix} X&Y&Z \end{pmatrix} = \begin{pmatrix} A&B&C\\A'&B'&C'\\A''&B''&C'' \end{pmatrix} \begin{pmatrix} \xi & \eta & \zeta \end{pmatrix}$
Which is great because if we actually expand both of the $abc$ and $\alpha \beta \gamma$ matrices into linear equations and then compare that with the 
$A B C$ matrix, we can calculate what the multiplication is. Or, in other words,
$\begin{pmatrix} a&b&c\\a'&b'&c'\\a''&b''&c'' \end{pmatrix} \begin{pmatrix} \alpha & \beta & \gamma \\ \alpha ' & \beta ' & \gamma ' \\ \alpha '' & \beta '' & \gamma '' \end{pmatrix} = \begin{pmatrix} A&B&C\\A'&B'&C'\\A''&B''&C'' \end{pmatrix}$

I won’t do that here, because it’s a $3 \times 3$ matrix which I think would be messy. Instead, let’s try it on a $2 \times 2$. Taking,
$\begin{pmatrix} X&Y \end{pmatrix} = \begin{pmatrix} a&b\\a'&b' \end{pmatrix} \begin{pmatrix} x&y \end{pmatrix} = \begin{pmatrix} (a,b)(x,y)&(a',b')(x,y,) \end{pmatrix}$
And,
$\begin{pmatrix} x&y \end{pmatrix} = \begin{pmatrix} \alpha & \beta  \\ \alpha ' & \beta ' \end{pmatrix} \begin{pmatrix} \xi & \eta \end{pmatrix}$
Which gives you,
$\begin{pmatrix} (a,b)(x,y)&(a',b')(x,y,) \end{pmatrix} = \begin{pmatrix} (a,b)(\alpha \xi+ \beta \eta, \alpha ' \xi + \beta ' \eta)&(a',b')(\alpha \xi'+ \beta \eta', \alpha ' \xi' + \beta ' \eta') \end{pmatrix}$
Or:
$\begin{pmatrix} a(\alpha \xi+ \beta \eta) +b (\alpha ' \xi + \beta ' \eta) & a'(\alpha \xi'+ \beta \eta') + b' (\alpha ' \xi' + \beta ' \eta') \end{pmatrix} $
$= \begin{pmatrix} (a\alpha + b\alpha', a\beta + b\beta')(\xi, \eta) & (a'\alpha +b'\alpha', a'\beta+b'\beta')(\xi, \eta)  \end{pmatrix}$
Which implies:
$\begin{pmatrix} a & b \\ a'&b' \end{pmatrix} \begin{pmatrix} \alpha & \beta \\ \alpha' & \beta' \end{pmatrix} = \begin{pmatrix} a\alpha +b\alpha' & a\beta +b\beta' \\ a'\alpha +b'\alpha' & a'\beta +b'\beta' \end{pmatrix}$
And that’s the matrix multiplication we already use!
This exactly follows in the Cayley’s example, too, as you’ll find in his book.[1]
A much more interesting question is whether this follows in two compound matrices. What if we took two compound matrices, and defined the second one to be factored into two more compound matrices; and then attempted to see how we could factor the original multiplication into the last compound matrix? It’s a mouthful, but it’s what we did in this answer except we introduce more terms in it.
Well, I tried it and it does work out in the same way to lead you to this convention again — in one of the methods. (There are two ways of defining a non-conventional "row $\cdot$ row" multiplication. And while one of them does give the same result as this, the other one is inconsistent with itself!)
[1]: Arthur Cayley's A Memoir of the Theory of Matrices http://scgroup.hpclab.ceid.upatras.gr/class/LAA/Cayley.pdf  (Royal Society Publishing: 2010)
