  In this paper, we show that the characterisation of all determinant
inequalities for $n \times n$ positive definite matrices is equivalent to
determining the smallest closed and convex cone containing all entropy
functions induced by $n$ scalar Gaussian random variables. We have obtained
inner and outer bounds on the cone by using representable functions and
entropic functions. In particular, these bounds are tight and explicit for $n
\le 3$, implying that determinant inequalities for $3 \times 3$ positive
definite matrices are completely characterized by Shannon-type information
inequalities.
