Both examination of a correlation matrix and a principal components analysis will provide useful information about the linear relationships between your predictors.
In broad terms:
- The correlation matrix will highlight bivariate relationships. It can detect pairs of variables with particularly high correlations where from a regression context you might want to remove one of the two or create a composite of the two.
- PCA will highlight multivariate relationships (e.g., sets of highly intercorrelated variables; the degree to which the set of variables can be effectively modelled by a smaller number of composites). It can be useful in informing the creation of composites of the variables.
Also, in broad terms there are differences based on whether you are taking a theoretical or a predictive orientation to your regression problem. If you have a more theoretical orientation, conceptual reasons may influence when and how you create composites and which if any correlated predictors you drop from an analysis. If you have a predictive orientation, you will be more concerned with maximising prediction, albeit preferably some form of out-of-sample prediction. If you have a theoretical orientation, then you are more likely going to be building arguments from an overall examination of the correlation matrix and the PCA analysis.