Matrix identities on weighted partial Motzkin paths

We give a combinatorial interpretation of a matrix identity on Catalan numbers and the sequence (1,4,42,43,…) which has been derived by Shapiro, Woan and Getu by using Riordan arrays. By giving a bijection between weighted partial Motzkin paths with an elevation line and weighted free Motzkin paths, we find a matrix identity on the number of weighted Motzkin paths and the sequence (1,k,k2,k3,…) for k≥2. By extending this argument to partial Motzkin paths with multiple elevation lines, we give a combinatorial proof of an identity recently obtained by Cameron and Nkwanta. A matrix identity on colored Dyck paths is also given, leading to a matrix identity for the sequence (1,t2+t,(t2+t)2,…).