This paper uses the combinatorics of Young tableaux to prove the plactic monoid of infinite rank does not satisfy a non-trivial identity, by showing that the plactic monoid of rank $n$ cannot satisfy a non-trivial identity of length less than or equal to $n$. A new identity is then proven to hold for the monoid of $n \times n$ upper-triangular tropical matrices. Finally, a straightforward embedding is exhibited of the plactic monoid of rank~$3$ into the direct product of two copies of the monoid of $3\times 3$ upper-triangular tropical matrices, giving a new proof that the plactic monoid of rank~$3$ satisfies a non-trivial identity.