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I'm quite confortable with definitions such as a derivative or an integral because i get to see why they are the way they are and that feels really natural, there is a fundamental question to be solved such as "What is the slope at a certain point of the curve" and so on... I can't see why someone would come up with the idea of a determinant, don't tell me that gives some area , I don't see why giving an area or a volume solves the kind of problems determinants do, is there a nice line of reasoning for determinants? Thank you guys !

Victor Luiz
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    You'll probably find [this question](https://math.stackexchange.com/q/668/81360) and its answers helpful. – Ben Grossmann Jul 12 '17 at 15:49
  • "Natural" is a rather subject term. Might be useful to add more information about what it really means. –  Jul 12 '17 at 15:55
  • I would like to know what was the fundamental problem one was trying to solve when came up with that tool. – Victor Luiz Jul 12 '17 at 16:00
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    Might be useful: [Historically, determinants were used long before matrices](https://en.wikipedia.org/wiki/Determinant#History) –  Jul 12 '17 at 16:03
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    They were originally used to tackle systems of linear equations. One of the first important discoveries, if the determinant is zero, the system does not have a singular solution. Cramer's rule is an important step along the way. The shoe-lace algorithm might be one of the early linkages of matrices to areas. – Doug M Jul 12 '17 at 16:13
  • "Don't tell me that and that" is a nice invitation to think about an appropriate answer to such a grand view question. – Christian Blatter Jul 12 '17 at 16:22
  • this is a nice bit of history http://www-groups.dcs.st-and.ac.uk/history/HistTopics/Matrices_and_determinants.html – Doug M Jul 12 '17 at 16:41

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