It depends on the criteria you have on "displayable".

If it's defined such that each edge-length has to be congruent to the distance in the matrix, you can simply follow the algorithm I sketched to see how each point's position is determined by the distance to 3 other points.

I.e. the distance to all other points can't be randomly chosen.

Since I had differential geometry at university I know that there are "distance-true"¹ projections originating from non-plane spaces like spheres².

This is no contradiction, since this data still has to fit into aforementioned algorithm.

The projection won't be "angle"¹ or "surface"¹-true, which is a problem for maps but not for graphs. I.a.W. the origin of the data doesn't need to be from a planar geometry but the graph needs to be planar euclidean!

I will try to update some WP links...

¹) not sure about the appropriate English terminology.

²) remembered it wrong sorry, see Map_projection#Metric_properties_of_maps, and in hindsight it's obvious that projections can only preserve distances if the Pythagorean theorem holds. Though preserving size of areas and angles are no problem.