A Liouville-Green (or WKB) asymptotic approximation theory is developed for the class of linear second-order matrix differential equations Y''= [f(t) A + G(t)] Y on [a, + infinity), where A and G(t) are matrices and f(t) is scalar. This includes the case of an ''asymptotically constant'' (not necessarily diagonalizable) coefficient, A (when f(t)=1). An explicit representation for a basis of the right-module of solutions is given, and precise computable bounds for the error terms are provided. The double asymptotic nature with respect to both, t and some parameter entering the matrix coefficient, is also shown. Several examples, some concerning semi-discretized wave and convection-diffusion equations, are given.