[Math] Why do determinants have their particular form

Question

I know that for a matrix $A$, if $\det(A)=0$ then the matrix does not have an inverse, and hence the associated system of equations does not have a unique solution. However, why do the determinant formulas have the form they do? Why all the complicated co-factor expansions and alternating signs ?

To sum it up: I know what determinants do, but its unclear to me why. Is there an intuitive explanation that can be attached to a co-factor expansion??..

Answer 1

Two exercises that may give you the answer you need (no work, no gain):

1. Assume you have a square $[0,1]\times [0,1]$ in the $(x,y)$-plane. Assume for some reason you need to change the variables you are using. The new variables you are using are now $w=a x + b y$ and $z=c x + d y$, where $a,b,c$ and $d$ are numbers. What is the area of the original square under the new coordinate system, the $(w,z)$-plane?
2. A multi-linear mapping in $\mathbb{R}^2$ (bilinear in this case) is a function, $M:\mathbb{R}^2\times \mathbb{R}^2\rightarrow \mathbb{R}$ such that $M( ax+b \hat x, y)= a M(x,y)+b M(\hat x,y)$ and $M(x,a y + b\hat y)=a M(x,y)+bM(x,\hat y)$. The map is alternating if $M(x,y)=-M(y,x)$. These two properties are very useful. Exercise: Show that if $M$ has these properties then $M(x,y)=k\cdot det\pmatrix{x_1 & y_1 \\ x_2 & y_2}$.