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What are matrices actually used for practically? (if at all)

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Solving systems of linear equations and systems that can be approximated to linear equations. Do you really need more detail than that?

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At the barest minimum, they are a very compact notation.

Sure, turning

$y_1 = a_{11}x_1 + a_{12}x_2 +b_1$

$y_2 = a_{21}x_1 + a_{22}x_2 +b_2$

into

$\mathbf{y}=\mathbf{A}\cdot\mathbf{x}+\mathbf{b}$ doesn't look like much savings, but what if there were 1000 variables?

I'd much rather write out $\mathbf{y}=\mathbf{A}\cdot\mathbf{x}+\mathbf{b}$ than 1000 equations with 1000 variables. And yes, people do work with 1000 variable systems. Computational fluid dynamics is one area where many thousands of equations are solved simultaneously.

In short, matrices are used quite a lot in mathematics. Studying them is well worth it.

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Well ,wherever you have many variables ,matrices is usually a very good way to go.For instance ,if you have a linear equations system ,with many many variables ,matrices give you the tools to study and solve them systematically and efficiently.But their field of applications ,does not stop there.The elements of a matrix can be vector coordinates ,so they are useful in some geometry applications too.They can relate to function variables ,polynomials etc.They are useful in handling problems in various fields of Mathematics ,with linear algebra methods.

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They are also useful as representations of things like groups and algebras.

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• 3 months later...
Matrices are used in just about every real world multidimensional problem!

Why are you posting in threads that haven't been posted in for several months? The other one you gave an answer to a problem that was asked back in October! I sincerely hope that the OPs have moved on in the several months since these threads have last been active, and I don't really know what you are trying to accomplish...

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