Please, I'm studying Matrix Algebra and am faced with the hard formulas for calculating a determinant of a Matrix. However, I really don't understand what does a determinant abstractly mean. I tried to search for a conceptual definition for the determinant in order to be able to understand and accept the idea of the determinant. But all my resources provide a functional definition.
For example; in Wikipedia, they say, it's a useful value that can be calculated by a square matrix. In an abstract algebra text, the author defines it via Leibniz's formula. In "The Theory of Matrices", prof. Matcher doesn't provide a definition at all.
What I'm looking for is: What does the determinant really mean? Why did they need to calculate it? Why did they calculate it that way and not in another way? Does the determinant provide any information about the equations that the matrix represents?
I hope I could explain what I mean... Thanks...