# A conceptual definition of a Matrix Determinant

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...

Turns out we can encapsulate these properties by saying that we want a multilinear (i.e. linear in each entry) alternating (i.e. switches sign if you swap entries once) function from $n$ vectors in $\mathbb{R}^n$ to $\mathbb{R}$ such that its value on the identity is 1. Interestingly, there is only one such function, and that is what we call the determinant. This method defines the determinant as an alternating tensor.
Alternative formulations include motivating it via group theory, by taking a sum over the symmetric group on $n$ elements of appropriate terms. You can also define it via a recursive formula which is what we tend to do in intro linear algebra courses. All these formulations are equivalent, which hints toward the fact that’s it’s something essential.