This article studies the constrained matrix-scaled consensus problem in networked systems, which is a novel generalization of conventional scaled consensus with scalar ratios. By introducing positive definite and negative definite scaling matrices, dynamical agents in a cooperative network are enabled to converge to equilibria regulated by the inverse of their individual scaling matrices. In our framework, the final states of agents can be related by a general matrix transformation accommodating the correlation between state components instead of just by a scalar multiplication in the conventional scaled consensus scenario. Moreover, we propose a gradient-based distributed protocol to guarantee that the states of all agents remain in their respective constraint sets. The analysis tool developed in this article is based on matrix analysis and Lyapunov stability theory. Numerical examples are provided to illustrate the effectiveness of the theoretical results. In particular, we show that the consensus-reaching process can be accelerated by utilizing a constraint set.