Representations of twisted tensor products

We obtain a faithful representation of the twisted tensor product $B{\otimes }_{\chi }A$ of unital associative algebras, when $B$ is finite dimensional. This generalizes the representations of [C] where $B=K[X]\mathrm{/}<X^2-X>$, [GGV] where $B=K[X]\mathrm{/}<X^n>$ and [JLNS] where $B=K^n$. Furthermore, we establish conditions to extend twisted tensor products $B{\otimes }_{\chi }A$ and $C{\otimes }_{\psi }A$ to a twisted tensor product $(B\times C){\otimes }_{\phi }A$.