This note shows that if $p>0$ and if $S_+$ is the set of symmetric positive definite matrices, then the function on $S_+$ defined by $A\mapsto \exp \smash {(-{\rm Trace}\,p\sqrt {2A})}$ is the Laplace transform of a non positive function concentrated on $S_+$ if $n\ge 2$. This function is explicitely computed for $n=2$. This computation is generalized to a Lorentz cone. The link of this question with the inverse Gaussian distributions in probability theory is also discussed, as well as the general problem of considering $\det L(A)$ as a Laplace transform on symmetric matrices when $L(\lambda )$ is a Laplace transform on the real line.