determinant and eigenvalues

For an nxn matrix A

det(A) = ∏(i=1, .., n) λi ...... [1],

where the λi are the eigen values (appearing with the correct multiplicities)

the characteristic equation is det(λI-A)=0, and this is an nth polynomial. Now i can see that statment [1] is true, but not sure how to demontrate this fact. We know that the matrix will be singular if an only if atleast one of the eigenvalues are zero, is there a way i can use this to show that [1] is true? need some advice on how to get started (∏ is from i=1 to n if it was not clear)