Calculate the absolute magnitude for a multi-star system We know that the absolute magnitude of a single star can be calculated with the formulae:
$$M_{\rm bol} = 4.75 -2.5 \log_{10} \left(\frac{L_*}{L_{\odot}}\right)\ ,$$
$$M = -2.5  \log_{10}(L_* / L_0),$$
where $L_*$ is the luminosity of the given star, and $L_0$ being the luminosity of a magnitude 0 star at a distance of 10 pc ($79L_\odot)$.
However, this only works for a single star, not multiple star systems. Is the formula for the absolute magnitude of a multiple star system defined as $L_*=L_1+L_2+...+L_n$ (the sum of all luminosities in the system), where $L_n$ is the luminosity of a star in the system, or is it something else?
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.