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Why would I want to use a matrix(2)? Previous explanations have stated where matrices are applied. That doesn't answer the question: what's the point of them, when surely you could just use algebraic expansions of them in the first place? It seems a strange way of representing basic algebraic expressions in a bizarre way to just to make things difficult. I've obviously missed something. Why use them? Why was this method of representing expessions devised? Why represent expessions in such an apparently unintuitive fashion?

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closed as not a real question by M Turgeon, Grigory M, Austin Mohr, Simon Markett, Mark Bennet Aug 30 '12 at 22:01

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

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I agree that they're confusing, and, to the uninitiated, not at all intuitive. But the power of mathematics is that it condenses complex concepts down to (relatively) concise expressions, and those concise expressions are often easier to manipulate/solve than the more complex concepts. It's definitely a trade-off -- a "deal with the devil" -- but it's allowed civilization to progress rapidly for at least 500 years. –  Daniel R Hicks Aug 30 '12 at 20:40
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"Previous explanations have stated where matrices are applied. That doesn't answer the question: what's the point of them"...Well, if the usages of a tool is not a good enough display of the tool's value, then probably nothing would convince you! Tools gain value from their utilization. If you don't want to use a tool, don't. Some people could have thought of the same thing about binary arithmetic until computers became common. Now, if computers don't convince someone of the value of the binary arithmetic, then we have a problem. –  Emmad Kareem Aug 30 '12 at 20:48
    
I think another way to phrase your question is "What is the intuition behind matrices?" –  rschwieb Aug 30 '12 at 20:49
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@JackManey: I don't think the comment "Who up-voted this garbage?" is appropriate anywhere on MSE. If you think the question deserves a downvote, then do so, and leave it at that. –  Eric Naslund Aug 30 '12 at 23:15
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@JackManey: I am not defending the question, how it is phrased, or what is being asked. The site has methods of dealing with such questions, and it was appropriately closed and downvoted. There is no need for comments saying "Who up-voted this garbage?" Such tone and disrespect has no place here. Period. –  Eric Naslund Aug 31 '12 at 2:56

3 Answers 3

Because they're an easy way to represent linear transformations on finite dimensional vector spaces.

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"The language of mathematics is unintelligible to the uninitiated." May be OP doesn't even know what a vector spaces is, or what a linear transformation is! –  user2468 Aug 30 '12 at 20:56
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@JackManey: This is your third comment that has been flagged by other users as rude or offensive. Don't misinterpret what I am saying, I think your answer is perfectly fine and entirely on topic for this question, and because of this I personally voted it up. However, comments of the form "if the OP stops throwing a whiny temper tantrum long enough to actually verify that such is the case..." are completely inappropriate, and don't belong in the MSE community. –  Eric Naslund Aug 30 '12 at 23:17
    
+1. I'd add "because they save a potload of writing out linear equations in all their bloody details." –  Rick Decker Aug 31 '12 at 1:11
    
Again, if the OP wishes to clarify that he/she doesn't understand something about this answer, then he/she is welcome to. Until then, why should I put forth any more effort than he/she has? –  user5137 Aug 31 '12 at 2:55

Matrices are useful in a number of theoretical aspects and as a computational aid.

Fixing a basis for a vector space, matrices are very convenient ways to represent linear transformations. Moreover, they are very useful for computing compositions of linear transformations (such as computing the product of matrices by the multiplying across row of the first matrix and down the column of the other matrix method).

Solving systems of linear equations are often very useful in numerous applied setting. There are many useful tricks such as row reduction to find solutions.

In a more theoretical setting, often you may want to determine if linear transformations are similar, i.e. given linear transformation $A$ and $B$ does there exists a $C$ such that $CAC^{-1} = B$. One possible way of doing this, rather than trying to find this $C$ used to conjugate, is to try to put these matrices representation into rational canonical form.

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It seems a strange way of representing basic algebraic expressions in a bizarre way to just to make things difficult

Do you really want to deal with expressions in $n^2$ variables?

By example: you have a graphs with $n$ vertices. Calculations of shortest paths (think: directions from Google Maps) and other quantities often use matrix multiplication, where the matrices represent the graph. Now. Do you want to deal with something like: $G^2$ or:

$$\begin{bmatrix} {g_{1,{1}}}^{2}+g_{1,{2}}g_{2,{1}}+g_{1,{3 }}g_{3,{1}}&g_{1,{1}}g_{1,{2}}+g_{1,{2}}g_{2,{2}}+g_{1,{3}}g_{3,{2}}&g _{1,{1}}g_{1,{3}}+g_{1,{2}}g_{2,{3}}+g_{1,{3}}g_{3,{3}} \\ g_{2,{1}}g_{1,{1}}+g_{2,{2}}g_{2,{1}}+g_{2,{3}}g_ {3,{1}}&g_{1,{2}}g_{2,{1}}+{g_{2,{2}}}^{2}+g_{2,{3}}g_{3,{2}}&g_{2,{1} }g_{1,{3}}+g_{2,{2}}g_{2,{3}}+g_{2,{3}}g_{3,{3}}\\ g _{3,{1}}g_{1,{1}}+g_{3,{2}}g_{2,{1}}+g_{3,{3}}g_{3,{1}}&g_{3,{1}}g_{1, {2}}+g_{3,{2}}g_{2,{2}}+g_{3,{3}}g_{3,{2}}&g_{1,{3}}g_{3,{1}}+g_{2,{3} }g_{3,{2}}+{g_{3,{3}}}^{2} \end{bmatrix} $$ This is just a $3\times 3$ matrix, raised to only power $2$.

You have families of $n^2$ variables related together in an obvious pattern: some sort of linearity. Do you want to ignore these properties and patterns and just deal with messy expressions of $n^2$ variables? Or, more elegantly, resort to an abstraction which replaces this mess. Moreover, matrices (considered as numbers in a system of matrices) turned out to form interesting properties: see matrix ring; not to mention vector spaces etc.

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