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We often learn in a standard linear algebra course that a determinant is a number associated with a square matrix. We can define the determinant also by saying that it is the sum of all the possible configurations picking an element from a matrix from different rows and different columns multiplied by (-1) or (1) according to the number inversions.

But how is this notion of a 'determinant' derived? What is a determinant, actually? I searched up the history of the determinant and it looks like it predates matrices. How did the modern definition of a determinant come about? Why do we need to multiply some terms of the determinant sum by (-1) based on the number of inversions? I just can't understand the motivation that created determinants. We can define determinants, and see their properties, but I want to understand how they were defined and why they were defined to get a better idea of their important and application.

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    the determinant defines volume in n-dimensions. Munkrese Analysis on Manifolds text has a nice discussion. I'm not well-versed in the history you seek so I leave it to others!2012-09-12
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    Related question http://math.stackexchange.com/questions/668/whats-an-intuitive-way-to-think-about-the-determinant/. See the answers given there.2012-09-12
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    Possible duplicate of http://math.stackexchange.com/questions/81521/development-of-the-idea-of-the-determinant.2013-10-21

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