Why does matrix arithmetic work?

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I know how to do matrix multiplication, cross, product, dot product, etc. But I don’t get why they work.

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What do you mean “Why do they work”?

Because they follow abstract arithmetic operations which are, as far as we know, consistent…

This question is pretty much the same as asking why are the rules of monopoly producing the game of monopoly… Because it’s been defined that way…

I’m not sure what can of response you’re expecting here… What would it look like if it didn’t work?

[Here’s a really good youtube series](https://www.youtube.com/watch?v=kjBOesZCoqc&list=PL0-GT3co4r2y2YErbmuJw2L5tW4Ew2O5B) that explains linear algebra.

It explains it better than I could, but the gist is that a matrix can be viewed as a way of transforming a vector and matrix multiplication describes the end result of applying multiple transformations.

Say you have a table like this:

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Number of tickets sold

Adults: 50 | Children: 85 | Seniors: 60 | Veterans: 45

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And another table like this:

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Price of tickets:

Adults: $10 | Children: $8 | Seniors: $6 | Veterans: $6

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And you need to find the total dollar amount of sales. It makes sense that you need to multiply each item in the first table by the respective items in the second table. Then add them all up.

That’s a matrix operation.

With larger matrices, you’re just doing more steps.

They “work” because we all agreed on what each operation means so we could do them consistently. It’s like PEDMAS…you could totally do SAMPED and it *would* work if everyone did the same thing. You can redefine matrix operations any way you want and, as long as everyone does it the same way, it will “work”.

“Under the hood”, matrix arithmetic is just a shortcut to a big pile of linear equations. A dot product, by convention/agreement, is the equivalent to “if I project one vector onto this other vector, how long is it?”. You can do that with geometry, you can do that with trig, you can do that with a system of linear equations, you can do that with a matrix. The matrix operation is just a particular format of the linear equations.

Likewise, cross product is “if I have two vectors, what’s the vector that’s perpendicular to both and as long as their product?”.

For all the matrix operations they work because we *defined* them to be, “If I assume the columns of the matrix are vector coordinates, how do I manipulate those values to get the result I want based on the definition of the operation?”

If we all agreed that the *rows* of the matrix were equivalent to the vectors, we could still define matrix operations, they’d just be mechanically different than the ones we currently use.