Suppose I am modeling the sales by week for all 50 states in the United States for a beverage company and I want to see the effect of marketing on sales. Would adding a fixed effect and a random interaction by state make sense mathematically? Or are there issues with matrix invertability or any other issues? I know random effects are typically added if the level is drawn at random from a population of levels for that factor in experimental design so want to know if it makes sense in this case. I would then derive the final coefficient of the marketing variable as the sum of the fixed effect coefficient and random effect coefficient.
Assuming that you have multiple or repeated measures/observations per state, then yes, it makes sense to fit random intercepts for state. You are correct that random effects are used when the observed levels of a factor are taken from a wider population, however that is not the only justification. If you have a large number of levels and there is correlation of observations within each level, then fitting random intercepts is usually a good idea otherwise results could be biased.
I would then derive the final coefficient of the marketing variable as the sum of the fixed effect coefficient and random effect coefficient.
This doesn't really make sense. You would only be interested in the fixed effect of the marketing variable. You also mentioned:
random interaction by state
This also doesn't quite make sense. The state variable would only enter the model as a random intercept (because it is a grouping factor). The model may look something like this:
sales ~ marketing + other_variables + (1 | state)
and, as mentioned, you interest would centre on the marketing variable. If you wished to allow the effect of marketing to vary by state, which is what I am guessing you meant by "random interaction by state" then you could fit random slopes for marketing:
sales ~ marketing + other_variables + (marketing | state)
In most software the random effects are assumed to be normally distributed with a mean of zero, so the overall effect of marketing will simply come from the fixed effect for it.