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 !
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2You'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

1Might be useful: [Historically, determinants were used long before matrices](https://en.wikipedia.org/wiki/Determinant#History) – Jul 12 '17 at 16:03

2They 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 shoelace 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://wwwgroups.dcs.stand.ac.uk/history/HistTopics/Matrices_and_determinants.html – Doug M Jul 12 '17 at 16:41