I always had some difficulties understanding the concept of determinant. Whenever I read about it, and that was usually from Linear Algebra books, it was always described with a lot of "mathematical formality" and abstractness, and although I could follow and understand the definition and properties, the concept generally seemed to me like coming out of the blue.
"The best intuition" I have about determinant is that it measures how much does space get scaled by certain transformation. This intuition helped me a lot to understand some other concepts involving determinant, for example deriving the proof and explaining to myself why non-trivial solutions of Homogeneous System of Linear Equations occur when determinant of a matrix of the corresponding transformation equals 0. But, although I can prove this intuition holds for 2-dimension case, I find no way to generalize it for \(n\) dimensions.
I feel that concept of determinant is really important to understand because it shows up in many branches of mathematics. I would really appreciate if someone could tell me a story about the determinant. I say story because it doesn't need to be highly mathematical and should aim to answer the following two questions:
Why did mathematicians introduced the concept of determinant? Why they needed it at the first place?
Based on that, how they derived the formula and generalized it for any number of dimensions?
Thanks in advance!