  Very recently planar collections of Feynman diagrams were proposed by Borges
and one of the authors as the natural generalization of Feynman diagrams for
the computation of $k=3$ biadjoint amplitudes. Planar collections are
one-dimensional arrays of metric trees satisfying an induced planarity and
compatibility condition. In this work we introduce planar matrices of Feynman
diagrams as the objects that compute $k=4$ biadjoint amplitudes. These are
symmetric matrices of metric trees satisfying compatibility conditions. We
introduce two notions of combinatorial bootstrap techniques for finding
collections from Feynman diagrams and matrices from collections. As
applications of the first, we find all $693$, $13\,612$, and $346\,710$
collections for $(k,n)=(3,7), (3,8),$ and $(3,9)$ respectively. As applications
of the second kind, we find all $90\, 608$ and $30\,659\,424$ planar matrices
that compute $(k,n)=(4,8)$ and $(4,9)$ biadjoint amplitudes respectively. As an
example of the evaluation of matrices of Feynman diagrams, we present the
complete form of the $(4,8)$ and $(4,9)$ biadjoint amplitudes. We also start
the study of higher dimensional arrays of Feynman diagrams, including the
combinatorial version of the duality between $(k,n)$ and $(n-k,n)$ objects.
