Calculate the absolute magnitude for a multi-star system
Yes, you are correct. The luminosities can be added. Luminosity is the amount of electromagnetic energy emitted per unit of time (measured in $ \textrm{J} \cdot \textrm{s}^{-1} $ or $\textrm{W}$). So if you have a multiple star system, the total amount of energy emitted (i.e. the total luminosity) is simply the sum of the energy emitted by each of its components.
This does not take into account any absorption of the radiation of one of the components by one of the others.
In the book by Jean Meeus [1], the formula for adding stellar magnitudes can be found in Chapter 56:
$$ m = -2.5 \log_{10} \sum_{i=1}^{n} 10^{-0.4 m_i} $$
This formula is for the apparent magnitude, but would also apply for the absolute magnitude, which, after all, is only the apparent magnitude at $10\,\textrm{pc}$ distance.
$$ M = -2.5 \log_{10} \sum_{i=1}^{n} 10^{-0.4 M_i} $$
[1] Jean Meeus, 1998, Chapter 56: Stellar Magnitudes, from Astronomical Algorithms, Willmann-Bell Inc. Second Edition.