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.

The determinant is the product of the eigenvalues of the matrix.  If the matrix is invertible, the eigenvalues of that inverse matrix are the reciprocals of the eigenvalues of the original matrix -- this fails if one of them is zero.  Hence,
($A$ is non-invertible) $\leftrightarrow$ ($A$ has a zero eigenvalue) $\leftrightarrow$ (det $A=0$).
