I understand that correlation (I'm referring to Pearson product-moment correlation coefficient) and regression share some similarities from this post: What's the difference between correlation and simple linear regression?
However, I'm wondering the limitations of multiple regression.
Scenario: I would like to explore the associations between a number of continuous variables (e.g., 10 variables). Of course, I would first compute a correlation matrix to observe their inter-correlations.
Next, I would like to explore the effect of the variables on the one or two DVs I’m interested in. I am wondering whether I should run many multiple regression models. I understand that even if two variables (X1,X2) have a significant correlation with another variable (Y), entering X1 into the regression model first might make X2 insignificant. It means that X2 being insignificant in the model doesn't mean X2 is useless. Therefore, my question is: in general, why do we make a regression model even if our correlation matrix could address some of my questions?