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Can anyone please explain how, why, and where determinants were developed/formalized? What was their historical usage? Why were they initially formulated and what were they used for (and later generalized for)?

I'm asking because I'm trying to truly understand the meaning of the determinant. I somewhat understand that it represents the distortion of volume of a region represented by a matrix after its transformation, but I find it hard to believe that that's the historical representation for it.

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  • $\begingroup$ I think all these linear algebra things came from solving system of linear equations. $\endgroup$ – peterwhy Aug 26 '13 at 16:10
  • $\begingroup$ In case you haven't looked yet, this is a good place to start $\endgroup$ – Omnomnomnom Aug 26 '13 at 16:26
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So I provided this link in the comment above, but I thought it might help to expand a little bit.

Determinants were originally considered as the property of a system of equations, beginning with Chinese mathematics in the $3^{rd}$ century BCE. That is, the determinant was thought of as a number associated with a system of linear equations; if that number is zero, then the system has no unique solution.

From there, determinants were thought of more in their own right, starting with Vandermonde in 1771. It's probably at this time that the geometric intuition behind it gained traction. Interestingly, the fact that $\det(AB)=\det(A)\det(B)$ was discovered before matrix multiplication was defined.

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The absolute value (modulus) might be the most origin of determinant from a cultural and rather trivial point of view, indeed there is a deep connect to the very roots of valuation (and norm). In that perspective the origin of determinant can be sealed and determined, somehow trivially, by history of human kind when value and norms found meaning in even primitive measurements.

BC $300-200$ there is evidence from Babylonians and Chinese that they applied equation-solving via semi-matrices but it is unclear whether an interpretation of determinants can be justified, and rather not explicit enough.

This is however rather too scrutinized history.

But more fixed on the term determinant in math discipline, indeed the first appearance of a determinant in Europe was in 1683 in a note by Leibniz to de l'Hôpital, explaining that the system of equations

$$10 + 11x + 12y = 0$$ $$20 + 21x + 22y = 0$$ $$30 + 31x + 32y = 0$$

has a solution because

$$10\cdot21\cdot32 + 11\cdot22\cdot30 + 12\cdot20\cdot31 = 10\cdot22\cdot31 + 11\cdot20\cdot32 + 12\cdot21\cdot30$$

this is exactly the condition that the coefficient matrix has determinant 0. Leibnitz does not use nummerical coefficients but two characters, the first marking in which equation it occurs, the second marking which letter it belongs to.

The intuition and reasoning of application of determinants was originally tied with equation solving.

By the way it was Laplace who tailored the term and used the word resultant for what we today call determinant and the term determinant was first introduced by Gauss in Disquisitiones arithmeticae (1801) while discussing quadratic forms (see also here>>>).

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