Can a correlation matrix furnish the information needed to conduct a path analysis? Yes.
Can simple inspection of correlations among variables unerringly identify a dependent variable? No; that depends on your research question(s), theory, or whatever rationale you have to support investigation of a specific path model.
As the name Path Analysis implies, you should be able to create a diagrammatic summary of the relationships between your independent and dependent variables. It is especially useful for describing mediation in terms of direct and indirect effects.