Intuitive understanding of determinants? 
For a $n \times n$ matrix $A$:$$\det (A) = \sum^{n}_{i=1}a_{1i}C_{1i}$$
where $C$ is the cofactor of $a_{1i}$. If the determinant is $0$, the matrix is not invertible.

Could someone an intuitive explanation of why a zero determinant means non-invertibility? I'm not looking for a proof, the book gives me one. I'm looking for intuition.
For example, consider the following properties of zero:
Property 1: If a factor of a polynomial is $0$ at $x$, then the polynomial is zero at $x$.
Intuition: Anything times zero, so if the remaining polynomial is being multiplied by zero, it has to be zero.
Property: $x + 0 = x$
Intuition: If you have something and don't change anything about it, it remains the same.
So why does having a zero determinant imply non-invertibility? Could someone give me a similar intuition for determinants?
Thanks.

There is a simple property for determinants:
$Det(A)Det(b)=Det(AB)$, so if you take:
$Det(A A^{-1})= Det(A)Det(A^{-1})= Det(I) = 1$, from this follows than $det(A)$ must be different from $0$ to be invertible.
