In this paper we establish relationships of the approximation numbers of matrix transformations acting between the vector-valued sequence spaces spaces of the type $\lambda(X)$ defined corresponding to a scalar-valued sequence space $\lambda$ and a Banach space $(X,\|.\|)$ as $$\lambda(X)=\{\overline x=\{x_i\}: x_i\in X, \forall~i\in \mathbb{N},~\{\|x_i\|_X\}\in \lambda\};$$ with those of their component operators. This study leads to a characterization of a diagonal operator to be approximable. Further, we compute the approximation numbers of inclusion maps acting between $\ell^p(X)$ spaces for $1\leq p\leq \infty$.