# Do you need to correct for a confounding factor if two groups are matched?

I'm looking for group differences in a variable (Power in a frequency band) that depends on age (i.e. increases linearly with age). If the two groups are matched for age, do I still need to include age in my model (e.g. Power ~ Group + Age)?

Also if the relationship of Power with Age differs between the two groups (e.g. the slope is steeper for Group 1 than for Group 2), should I include an interaction term in my model to take it into account (e.g. Power ~ Group * Age)?

I guess I could generalize the second part by asking when do we need to include an interaction term in a linear model to correct for a confounding factor?