After much help from this community, I finally have a way to plot the real solutions to the determinant. However, this may seem like a stupid request, I want to combine both figures that result from this code into one with a legend for each w.
k = 9.; l = 12.; m = 2.; M = 4.;
mat = {{m*w^2 - 2*k, k, k*zeta}, {k, M*w^2 - (l + k),
l}, {-k*zeta, l, M*w^2 - (k - l)}};
mydet = ExpToTrig[Det[mat]];
sol = Solve[mydet == 0, w];
funcs = Chop[w /. sol /. {zeta -> Exp[-3I ka]} // FullSimplify];
Show[Table[Plot[Max[0, (funcs[[k]] + Conjugate[funcs[[k]]])/2], {ka, 0, \[Pi]}, PlotStyle -> Black], {k, 1, Length[funcs]}], PlotRange -> All]
Show[Table[Plot[Max[0, (funcs[[k]] - Conjugate[funcs[[k]]])/(2 I)], {ka,
0, \[Pi]}, PlotStyle -> Blue], {k, 1, Length[funcs]}], PlotRange -> All]
I hope someone can answer this so I can finally be done with this project.