Symbolic matrix systems are generalizations of finite symbolic matrices for sofic systems to subshifts. We prove that if two symbolic matrix systems are strong shift equivalent, then the gauge actions of the associated C*-algebras are stably outer conjugate. The proof given here is based on the construction of an imprimitivity bimodule from a bipartite $\lambda$-graph system, so that an equivariant version of the Brown–Green–Rieffel Theorem proved by Combes is used, together with its proof. As a corollary, if two subshifts are topologically conjugate, then the gauge actions of the associated C*-algebras are stably outer conjugate.