  We present a random matrix interpretation of the distribution functions which
have appeared in the study of the one-dimensional polynuclear growth (PNG)
model with external sources. It is shown that the distribution, GOE$^2$, which
is defined as the square of the GOE Tracy-Widom distribution, can be obtained
as the scaled largest eigenvalue distribution of a special case of a random
matrix model with a deterministic source, which have been studied in a
different context previously. Compared to the original interpretation of the
GOE$^2$ as ``the square of GOE'', ours has an advantage that it can also
describe the transition from the GUE Tracy-Widom distribution to the GOE$^2$.
  We further demonstrate that our random matrix interpretation can be obtained
naturally by noting the similarity of the topology between a certain
non-colliding Brownian motion model and the multi-layer PNG model with an
external source. This provides us with a multi-matrix model interpretation of
the multi-point height distributions of the PNG model with an external source.
