Why is the formula for the determinant as it is? I know this sounds like such a vague question, but seeing as we all know the formula for the determinant of a general nxn matrix, I want to know exactly why we define it as such. I know that determinants are used to define whether or not a matrix has a unique solution, but surely there much be other methods and other equations that we can use to determine this. So again, why is it that we use the current equation and the current definition?
 A: Historically the determinant was introduced by Gerolamo Cardano as an expression to determine if a linear $2\times 2$ system has a solution. Later this was generalized by Leibniz for larger linear systems. 
You are right if you ask, why we use this particular notion in order to determine if a linear system has a solution or not. It is possible to define a different function that is zero if and only if the matrix induces a linear system that has a solution (say you can multiply the determinant with a scalar, or you take the absolute value). However, the determinant has other nice geometrically properties, which are helpful in many different applications. 
Here is just one example. A $n\times n$ matrix $T$ decodes a linear transformation $t \colon \mathbb{R}^n \to \mathbb{R}^n$. If you consider the volume of a $n$-simplex and its image in $t$, then the relative change of the simplex volume is $\det(T)$. In particular, if $\det(T)<0$ then the orientation of the simplex in the image is flipped. 
So it makes sense to define the determinant this way, because it comes in handy for many applications.
