Determinant of a particular symmetric matrix

Hi all,
My problem is the following.
I have a symmetric n x n matrix, which contains only positive values.
Each value on the main diagonal is equal to 1, and other values v are such that 0 < v < 1.
Numerically, it seems that the determinant of such a matrix
is always different from 0, so that it always has an inverse (very interesting to me).
I'd like to know if there exist a proof of this property (if it's true).
If someone could help me...
thanks.
.