  We give a combinatorial interpretation of a matrix identity on Catalan
numbers and the sequence $(1, 4, 4^2, 4^3, ...)$ 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 $(1, k, k^2, k^3, ...)$ for any $k \geq 2$. 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 $(1, t^2+t, (t^2+t)^2, ...)$.
