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I learned linear algebra from books of Friedberg, Gilbert & Strang, Anton etc. by myself. I dare say, that I learned all that stuff eagerly.
Studying by myself, I could not intuitively understand the definition of the determinant (its even – odd manner). I could only memorize the definition, and then use it (or try to use it) to solve some related readers' homework problems or other exercises.
As you know, the purpose of a determinant is literally to determine whether a given system of equations has a unique solution or not.
In other words, the "determinant" will determine whether the row vectors (and equivalently, column vectors) of a given square matrix are independent or not.
If those are mutually independent, then they can geometrically represent an $n$-dimensional quantity (for example, area in 2 dimensions or volume in 3D). If not, some of them are dependent, so they cannot form the $n$-dimensional quantity, and correspondingly the determinant is zero.
A multiple of any row can be added to another, this kind of row elementary operation does not change the determinant value. The picture below illustrates an intuitive understanding of that, too.
Writing the sides of the parallelogram as rows or columns of a square matrix, this transformation transforms it to another with the same value of the determinant.
It can be transformed to Gauss–Jordan form, in this case, each of the row / column vectors are orthogonal because their inner products are all zero. (I tried this with the Gram–Schmidt process; however, intuitively, the result is surely the same.)
Those vectors are orthogonal so it is very clear that just multiplication of the diagonal terms should give directly the aforementioned $n$-dimensional quantity, so that's the determinant in such case.
I understand the determinant in this manner, and it makes sense intuitively.
However the textbook definition mentioned above (defined in the "even–odd" manner) looks very weird to me.
What is the motivation of that definition? And can it be generally derived from my intuition about the $n$-dimensional quantities? I succeeded in doing so for the $2\times2$ and $3\times3$ cases, but I cannot see any generalized relation.
It seems to me that the definition of the determinant comes down magically, without enough logic.
I was wondering if you could help me.
Thank you in advance.