  We provide a decomposition that is sufficient in showing when a symmetric
tridiagonal matrix $A$ is completely positive. Our decomposition can be applied
to a wide range of matrices. We give alternate proofs for a number of related
results found in the literature in a simple, straightforward manner. We show
that the cp-rank of any irreducible tridiagonal doubly stochastic matrix is
equal to its rank. We then consider symmetric pentadiagonal matrices, proving
some analogous results, and providing two different decompositions sufficient
for complete positivity. We illustrate our constructions with a number of
examples.
